diff --git a/nonlinear.pdf b/nonlinear.pdf index 622a838..1f543fd 100644 Binary files a/nonlinear.pdf and b/nonlinear.pdf differ diff --git a/nonlinear.tex b/nonlinear.tex index 7410d5c..887e4ca 100644 --- a/nonlinear.tex +++ b/nonlinear.tex @@ -1,1980 +1,1825 @@ %%%%%%%%%%%%%%%%%%%%%%% file template.tex %%%%%%%%%%%%%%%%%%%%%%%%% % % This is a general template file for the LaTeX package SVJour3 % for Springer journals. Springer Heidelberg 2010/09/16 % % Copy it to a new file with a new name and use it as the basis % for your article. Delete % signs as needed. % % This template includes a few options for different layouts and % content for various journals. Please consult a previous issue of % your journal as needed. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % First comes an example EPS file -- just ignore it and % proceed on the \documentclass line % your LaTeX will extract the file if required \begin{filecontents*}{example.eps} %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 19 19 221 221 %%CreationDate: Mon Sep 29 1997 %%Creator: programmed by hand (JK) %%EndComments gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore \end{filecontents*} % \RequirePackage{fix-cm} % %\documentclass{svjour3} % onecolumn (standard format) %\documentclass[smallcondensed]{svjour3} % onecolumn (ditto) \documentclass[smallextended]{svjour3} % onecolumn (second format) %\documentclass[twocolumn]{svjour3} % twocolumn % \smartqed % flush right qed marks, e.g. at end of proof % \usepackage{amsmath} \usepackage{amssymb} \usepackage{graphicx} \usepackage{algorithm} \usepackage{color} \usepackage{hyperref} \hypersetup{ colorlinks=true,% citecolor=black,% filecolor=black,% linkcolor=black,% urlcolor=blue } % 1 inch margins when printed from ps \hoffset=0in \voffset=0in \evensidemargin=0in \oddsidemargin=0in \textwidth=6.5in \topmargin=0in \headheight=0.0in \headsep=.5in \textheight=8in % % \usepackage{mathptmx} % use Times fonts if available on your TeX system % insert here the call for the packages your document requires %\usepackage{latexsym} % etc. \PassOptionsToPackage{normalem}{ulem} \newcommand{\minimize}[2]{\ensuremath{\underset{\substack{{#1}}}% {\mathrm{minimize}}\;\;#2 }} \newcommand{\Argmind}[2]{\ensuremath{\underset{\substack{{#1}}}% {\mathrm{Argmin}}\;\;#2 }} \newcommand{\Frac}[2]{\displaystyle{\frac{#1}{#2}}} \newcommand{\menge}[2]{\big\{{#1} \;|\; {#2}\big\}} \newcommand{\Pair}[2]{{\big\langle{{#1},{#2}}\big\rangle}} \newcommand{\pair}[2]{{\langle{{#1},{#2}}\rangle}} \newcommand{\Scal}[2]{{\bigg\langle{{#1}\:\bigg |~{#2}}\bigg\rangle}} \newcommand{\Menge}[2]{\bigg\{{#1}~\bigg|~{#2}\bigg\}} \newcommand{\lev}[1]{\ensuremath{\mathrm{lev}_{\leq #1}\:}} \newcommand{\enve}[1]{\ensuremath{\operatorname{env}#1}} \newcommand{\emp}{\ensuremath{{\varnothing}}} \newcommand{\bdry}{\ensuremath{\operatorname{bdry}}} \newcommand{\yosi}[2]{\ensuremath{\sideset{^{#2}}{}{\operatorname{}}\!\!#1}} \newcommand{\infconv}{\ensuremath{\mbox{\small$\,\square\,$}}} \newcommand{\scal}[2]{\left\langle{#1}\mid {#2} \right\rangle} \newcommand{\pscal}[2]{\langle\langle{#1}\mid{#2}\rangle\rangle} \newcommand{\norm}[1]{\|#1\|} \newcommand{\vuo}{\ensuremath{\mbox{\footnotesize$\square$}}} \newcommand{\exi}{\ensuremath{\exists\,}} \newcommand{\zeroun}{\ensuremath{\left]0,1\right[}} \newcommand{\HH}{\ensuremath{\mathcal H}} \newcommand{\GG}{\ensuremath{\mathcal G}} \newcommand{\YY}{\ensuremath{\mathcal Y}} \newcommand{\XX}{\ensuremath{\mathcal X}} \newcommand{\bary}{\ensuremath{\widetilde{\boldsymbol{y}}}} \newcommand{\barp}{\ensuremath{\widetilde{\boldsymbol{p}}}} \newcommand{\barq}{\ensuremath{\widetilde{\boldsymbol{q}}}} \newcommand{\barx}{\ensuremath{\widetilde{\boldsymbol{x}}}} \newcommand{\BL}{\ensuremath{\EuScript B}\,} \newcommand{\BP}{\ensuremath{\EuScript P}} \newcommand{\HHH}{\ensuremath{\boldsymbol{\mathcal H}}} \newcommand{\WW}{\ensuremath{\boldsymbol{\mathcal W}}} \newcommand{\KKK}{\ensuremath{\boldsymbol{\mathcal K}}} \newcommand{\GGG}{\ensuremath{\boldsymbol{\mathcal G}}} \newcommand{\VV}{\ensuremath{\boldsymbol{V}}} \newcommand{\CCC}{\ensuremath{\boldsymbol{C}}} \newcommand{\MM}{\ensuremath{\boldsymbol{M}}} \newcommand{\SG}{\ensuremath{\boldsymbol{S}}} \newcommand{\EE}{\ensuremath{\boldsymbol{E}}} \newcommand{\XP}{\ensuremath{\mathcal X}^*} \newcommand{\sri}{\ensuremath{\operatorname{sri}}} \newcommand{\ri}{\ensuremath{\operatorname{ri}\,}} \newcommand{\RR}{\ensuremath{\mathbb R}} \newcommand{\KK}{\ensuremath{\mathcal K}} \newcommand{\RP}{\ensuremath{\left[0,+\infty\right[}} \newcommand{\RPP}{\ensuremath{\,\left]0,+\infty\right[}} \newcommand{\RX}{\ensuremath{\,\left]-\infty,+\infty\right]}} \newcommand{\NN}{\ensuremath{\mathbb N}} \newcommand{\SL}{\ensuremath{\EuScript S}\,} \newcommand{\dom}{\ensuremath{\operatorname{dom}}} \newcommand{\cont}{\ensuremath{\operatorname{cont}}} %\newcommand{\gr}{\ensuremath{\operatorname{gra}}} \newcommand{\prox}{\ensuremath{\operatorname{prox}}} \newcommand{\Prox}{\ensuremath{\operatorname{Prox}}} \newcommand{\intdom}{\ensuremath{\operatorname{int}\operatorname{dom}}\,} \newcommand{\inte}{\ensuremath{\operatorname{int}}} \newcommand{\cart}{\ensuremath{\mbox{\huge{$\times$}}}} \newcommand{\scart}{\ensuremath{\mbox{\LARGE{$\times$}}}} \newcommand{\WC}{\ensuremath{{\mathfrak W}}} \newcommand{\TT}{\ensuremath{{\mathbf T}}} \newcommand{\SC}{\ensuremath{{\mathfrak S}}} \newcommand{\rh}{\ensuremath{{\mathrm a}}} \newcommand{\og}{\ensuremath{{\mathrm b}}} \newcommand{\rk}{\ensuremath{{\mathsf {\mathbf a}}}} \newcommand{\ck}{\ensuremath{{\mathsf {\mathbf u}}}} \newcommand{\xk}{\ensuremath{{\mathsf{\mathbf x}}}} \newcommand{\yk}{\ensuremath{{\mathsf{\mathbf y}}}} \newcommand{\RPX}{\ensuremath{{[0,\pinf]}}} \newcommand{\card}{\ensuremath{\operatorname{card}}} \newcommand{\bd}{\ensuremath{\operatorname{bdry}}} \newcommand{\Argmin}{\ensuremath{\operatorname{Argmin}}} \newcommand{\argmin}{\ensuremath{\operatorname{argmin}}} \newcommand{\ran}{\ensuremath{\operatorname{ran}}} \newcommand{\zer}{\ensuremath{\operatorname{zer}}} \newcommand{\gra}{\ensuremath{\operatorname{gra}}} \newcommand{\conv}{\ensuremath{\operatorname{conv}}} \newcommand{\vv}{\ensuremath{\boldsymbol{v}}} \newcommand{\sss}{\ensuremath{\boldsymbol{s}}} \newcommand{\xx}{\ensuremath{\boldsymbol{x}}} \newcommand{\xs}{\ensuremath{\textsf{x}}} \newcommand{\xo}{\ensuremath{\overline{\boldsymbol{x}}}} \newcommand{\pp}{\ensuremath{\boldsymbol{p}}} \newcommand{\qq}{\ensuremath{\boldsymbol{q}}} \newcommand{\yy}{\ensuremath{\boldsymbol{y}}} \newcommand{\ff}{\ensuremath{\boldsymbol{f}}} \newcommand{\hh}{\ensuremath{\boldsymbol{h}}} \newcommand{\ttt}{\ensuremath{\boldsymbol{t}}} \newcommand{\ee}{\ensuremath{\boldsymbol{e}}} \newcommand{\rr}{\ensuremath{\boldsymbol{r}}} \newcommand{\gggg}{\ensuremath{\boldsymbol{g}}} \newcommand{\zz}{\ensuremath{\boldsymbol{z}}} \newcommand{\bb}{\ensuremath{\boldsymbol{b}}} \newcommand{\uu}{\ensuremath{\boldsymbol{u}}} \newcommand{\cc}{\ensuremath{\boldsymbol{c}}} \newcommand{\dd}{\ensuremath{\boldsymbol{d}}} \newcommand{\aaa}{\ensuremath{\boldsymbol{a}}} \newcommand{\ww}{\ensuremath{\boldsymbol{w}}} \newcommand{\BB}{\ensuremath{\boldsymbol{B}}} \newcommand{\LL}{\ensuremath{\boldsymbol{L}}} \newcommand{\PPP}{\ensuremath{\boldsymbol{\mathsf{P}}}} \newcommand{\UU}{\ensuremath{\boldsymbol{U}}} \newcommand{\EEE}{\ensuremath{\boldsymbol{E}}} \newcommand{\E}{\ensuremath{\mathbf{E}}} \newcommand{\D}{\ensuremath{\mathbf{D}}} \newcommand{\ep}{\ensuremath{\boldsymbol{\varepsilon}}} \newcommand{\RRR}{\ensuremath{\boldsymbol{R}}} %\newcommand{\RRR}{\ensuremath{\boldsymbol{R}}} \newcommand{\AAA}{\ensuremath{\boldsymbol{A}}} \newcommand{\BBB}{\ensuremath{\boldsymbol{B}}} \newcommand{\QQ}{\ensuremath{\boldsymbol{Q}}} \newcommand{\SSS}{\ensuremath{\boldsymbol{S}}} \newcommand{\DD}{\ensuremath{\boldsymbol{D}}} \newcommand{\PP}{\ensuremath{\boldsymbol{P}}} \newcommand{\FF}{\ensuremath{\boldsymbol{\mathcal{F}}}} \newcommand{\cone}{\ensuremath{\operatorname{cone}}} \newcommand{\Fix}{\ensuremath{\operatorname{Fix}}} \newcommand{\Id}{\ensuremath{\operatorname{Id}}} \newcommand{\diam}{\ensuremath{\operatorname{diam}}} \newcommand{\IId}{\ensuremath{\boldsymbol{\operatorname{Id}}}} \newcommand{\weakly}{\ensuremath{\rightharpoonup}} \newcommand{\minf}{\ensuremath{-\infty}} \newcommand{\pinf}{\ensuremath{+\infty}} \newcommand{\LLambda}{\ensuremath{\boldsymbol{\Lambda}}} \newcommand{\vva}{\ensuremath{\boldsymbol{\epsilon}}} \newcommand{\trace}{\operatorname{tr}} % AE's commands \newcommand{\edita}[1]{{\color{blue} #1}} \newcommand{\notea}[1]{{\color{magenta} \textbf{Note: #1}}} \newcommand{\ol}{\overline} \newcommand{\Null}{\operatorname{null}} \newcommand{\relint}{\operatorname{relint}} \newcommand{\row}{\operatorname{row}} % please place your own definitions here and don't use \def but % \newcommand{}{} % % Insert the name of "your journal" with % \journalname{myjournal} % \begin{document} \title{A relaxation of the augmented Lagrange method } %\subtitle{Do you have a subtitle?\\ If so, write it here} %\titlerunning{Short form of title} % if too long for running head \author{ %C. Vu\and Alcaoglu Ahmet\and Sahin M. Fatih\and Alp Yurtsever\and Volkan Cevher \\[5mm] } %\authorrunning{Short form of author list} % if too long for running head %\institute{Laboratory for Information and Inference Systems (LIONS), EPFL, Switzerland\\ % \email{bang.vu@epfl.ch\and ahmet.alacaoglu@epfl.ch; mehmet.sahin@epfl.ch;alp.yurtsever@epfl.ch; mehmet.sahin@epfl.ch;volkan.cehver@epfl.ch} } \date{Received: date / Accepted: date} % The correct dates will be entered by the editor \maketitle \begin{abstract} We propose a splitting method for solving .... \keywords{Non-linear constraint \and Non-convex \and Smoothing\and Primal-dual} % \PACS{PACS code1 \and PACS code2 \and more} \subclass{47H05\and 49M29\and 49M27\and 90C25} \end{abstract} \section{Introduction \label{intro}} \edita{Various problems in engineering and computational sciences can be cast as non-linear optimization programs, and the design of efficient numerical algorithms to provably solve such problems is therefore of fundamental importance. cite? %Non-linear programming is a broad discipline in applied mathematics. In this paper, we are particularly interested in solving the optimization program \begin{equation} \label{prob:01} \begin{cases} \min_{u} h(u),\\ A(u) = b,\\ u\in C, \end{cases} \end{equation} where $h:\mathbb{R}^d\rightarrow\mathbb{R}$ and $A:\mathbb{R}^d\rightarrow\mathbb{R}^m$ satisfy \begin{align} \| \nabla h(u) - \nabla h(u')\| \le \lambda_h \|u-u'\|, \qquad \| Dh(u) - Dh(u') \| \le \lambda_A \|u-u'\|, \label{eq:smoothness basic} \end{align} for every $u,u'\in \mathbb{R}^d$. Above, $\nabla h(u)\in \mathbb{R}^d$ is the gradient of $h$ and $DA(u)\in \mathbb{R}^{m\times d}$ is the Jacobian of $A$. %where $h\in \mathbb{L}_{d}(\lambda_h)$ is a continuously-differentiable function from $\mathbb{R}^d$ to $\RR$ with $\lambda_h$ Lipschitz-continuous gradient, $A: \mathbb{R}^d\rightarrow\mathbb{R}^m$ is a non-linear function, with each component $A_i\in \mathbb{L}_d(\lambda_A) $ for $i\in \{1,\cdots,m\}$. Moreover, $C\subset \RR^d$ is non-empty, closed, and convex. %\edita{Program \eqref{prob:01} is typically} non-convex and \edita{can be considered as a standard non-linear program, where the set $C$ %is modeled by inequalities constraints, namely, $C = \menge{u}{g(u) \leq 0}$ for some $g\colon\RR^d\to \RR^s$.} Variants of Program~\eqref{prob:01} naturally arise in a broad range of applications in ?? \notea{Please add some representative applications above alongside some references. } For the sake of brevity, we showcase here one instance of Program $\eqref{prob:01}$.} \begin{example}\edita{{{\textbf{(Burer-Monteiro factorization)}}} Let $\mathbb{S}^{d'\times d'}$ be the space of $d'\times d'$ symmetric matrices, equipped with the standard inner product $\langle x|y\rangle = \text{tr}(x^*y)$. In particular, when $x\in \mathbb{S}^{d'\times d'}$ is positive semi-definite, we write that $x\succeq 0$.} % Consider $\mathcal{C}'\subseteq \mathcal{X}$, and let $h_0\colon\mathcal{X}\to \RR$ be a differentiable %convex function, with $\mathbb{L}_{0}$ Lipschitz-continuous gradient.} % \edita{Consider the program \begin{equation} \label{e:fac} \begin{cases} \min_x h'(x) \\ A'(x) = b'\\ x\in C'\\ x \succeq 0 , \end{cases} \end{equation} where $h': \mathbb{S}^{d'\times d'} \to \RR$, $A'\colon\mathbb{S}^{d'\times d'}\to\RR^m$, $b\in\RR^m$, and $C' \subseteq \mathbb{R}^{d'\times d'}$. Variants of Program \eqref{e:fac} are popular in matrix completion and sensing \cite{park2016provable}, with a broad range of applications to problems in collaborative filtering, geophysics, and imaging, among others~\cite{Burer2005,Burer2003,tu2014practical}. Two common choices for $C'$ in Program \eqref{e:fac} are $C' =\{x: x \ge 0\}$ and $C' = \{x: \text{tr}(x) \le 1\}$ \cite{mixon2016clustering}.} \edita{Solving Program \eqref{e:fac} with semi-definite programming is not scalable, becoming increasingly cumbersome as the dimension $d'$ grows. To overcome this {computational bottleneck}, the factorized technique sets $x = uu^\top$ for $u\in \mathbb{R}^{d'\times r}$ and a sufficiently large $r$. The resulting non-convex program is then solved with respect to the much lower-dimensional variable $u$. If we also replace the constraint $uu^\top \in C'$ with $u\in C$ for a properly chosen convex set, the new problem in $u$ matches Program \eqref{prob:01} with $h(u) = h'(uu^\top)$ and $A(u) = A'(uu^\top)$. For our examples of $C'$ above, we might choose $C=\{u:u\ge 0\}$ and $C=\{\|u\|_F^2 \le 1\}$, respectively. Here, $\|\cdot\|_F$ stands for the Frobenius norm. } \end{example} \edita{The \emph{augmented Lagrangian method} \cite{luenberger1984linear} is a powerful approach to solve Program \eqref{prob:01}, see Section \ref{sec:related work} for a review of the related literature as well as other approaches to solve Program \eqref{prob:01}}. \edita{ Indeed, for positive $\beta$, it is easy to verify that Program \eqref{prob:01} is equivalent to \begin{align} \min_{u\in C} \max_y \, \mathcal{L}_\beta(u,y), \label{eq:minmax} \end{align} where \begin{align} \label{eq:Lagrangian} \mathcal{L}_\beta(u,y) := h(u) + \langle A(u)-b, y \rangle + \frac{\|A(u)-b\|^2}{2\beta}, \end{align} is the augmented Lagrangian corresponding to Program \eqref{prob:01}. The equivalent formulation in Program \eqref{eq:minmax} naturally suggests the following algorithm to solve Program \eqref{prob:01}:} \begin{equation}\label{e:exac} u_{k+1} \in \underset{u\in C}{\argmin} \, \mathcal{L}_{\beta}(u,y_k), \end{equation} \begin{equation} y_{k+1} = y_k+\frac{A(u_{k+1})-b}{\beta}. \end{equation} In fact, when the penalty parameter $\beta$ is sufficiently small, the augmented Lagrangian has a local minimum point near the true optimal point. However, we do not know exactly how small $\beta$ is. Hence, the choice of $\beta$ plays a centreral role in practices. \notea{Is the last claim really true? Programs \eqref{prob:01} and \eqref{eq:minmax} seem to be equivalent. } \edita{In our nonlinear framework, updating $u$ in the augmented Lagrangian method requires solving the non-convex Program \eqref{e:exac} to global optimality, which is often intractable. } \notea{We should discuss fixes to this issue, if any, and explain why they are not satisfactory.} \edita{The key contribution of this paper is to provably and efficiently address this challenge.} \edita{ \paragraph{\emph{\textbf{Contributions.}} } In order to solve Program \eqref{prob:01}, this paper proposes to replace the (intractable) Program \eqref{e:exac} with the update \begin{equation} u_{k+1} = P_C (u_k - \gamma_k \nabla \mathcal{L}_{\beta_k} (u_k,y_k)), \label{eq:new update} \end{equation} for carefully selected sequences $\{\beta_k,\gamma_k\}_k$. Here, $P_C$ is the orthogonal projection onto the convex set $C$ which is often easy to compute in various applications and consequently the update in \eqref{eq:new update} is inexpensive and fast. Put differently, instead of fully solving Program \eqref{e:exac}, this paper proposes to apply one iteration of the projected gradient algorithm for every update. We provide the convergence guarantees for this fast and scalable new algorithm. }\notea{We should summarize the guarantees.} %%%%%%%%%%%%%%%%%%%%% \section{ Preliminaries} \notea{I think the whole of this section should move down. The actual results are hidden deep in the paper!} \paragraph{\textbf{\emph{Notation.}}} We use the notations $\scal{\cdot}{\cdot}$ and $\|\cdot\|$ for the \edita{standard inner} product and \edita{the} associated norm on $\RR^d$\edita{, respectively}. \edita{The adjoint of \edita{a} linear operator is denoted the superscript $\top$.} Let $C\subset \mathbb{R}^d$ be nonempty, closed, \edita{and convex}. The indicator function of $C$ is denoted by $\iota_{\mathcal{C}}$, and the projection onto $C$ is denoted by $P_C$. %The distance function is $d_{\mathcal{C}}\colon u\mapsto \inf_{a\in\mathcal{C}}\|u-a\|$. The projection of $x$ onto $C$ is denoted by $P_Cx$. \edita{For $u\in C$, the tangent cone to $C$ at $u$ is \begin{equation} T_{C}(u) = \left\{v\in \RR^d : \exists t > 0 \text{ such that } u+t v \in C\right\}. \end{equation} The corresponding normal cone $N_C(u)$ at $u$ is the polar of the tangent cone, namely, \begin{align} N_C(u) = \left\{ v': \langle v, v' \rangle \le 0,\, \forall v\in T_C(u) \right\}. \end{align}} %The regular normal cone $\hat{N}_C(\overline{u})$ is defined as the dual to the tangent cone, $\hat{N}_C(\overline{u}) = T_{C}(\overline{u})^*$. %The (Mordukhovich) limiting normal cone to $C$ at $\overline{u}$ is defined by %\begin{equation} %N_C(\overline{u}) = \menge{v\in \RR^d}{\exists u_k\to \overline{u}, v_k\to v \;\text{with}\; (\forall k\in\NN)\; v_k \in \hat{N}_{C}(u_k)}. %\end{equation} \edita{ The sub-differential of a convex function $f$ at $u$ is defined as % %Let $f\colon \RR^d\to (-\infty, +\infty$ be a proper, lower semi-continuous, convex function. The sub-differential of $f$ at $p$ is \begin{equation} \partial f(u)= \left\{ g : f(u') - f(u) \ge \langle g, u'-u\rangle, \,\, \forall u' \right\}. \end{equation} } \edita{In particular,} if $f$ is differentiable at $u$, $\partial f(u)$ is a singleton and denoted by $\nabla f(u)$. \label{s:nota} \paragraph{\textbf{\emph{\edita{Necessary Optimality Conditions.}}} \label{sec:opt cnds}} \edita{Necessary optimality conditions} for \edita{Program} \eqref{prob:01} are well studied in the literature \cite[Corollary 6.15]{rockafellar2009variational}. \edita{Indeed, $u$ is a (first-order) stationary point of Program \eqref{prob:01} if there exists $y$ for which \begin{align} \begin{cases} -\nabla h(u) - DA(u)^\top y \in N_C(u)\\ A(u) = b. \end{cases} \label{e:inclu1} \end{align} Here, $DA(u)$ is the Jacobian of $A$ at $u$. Recalling \eqref{eq:Lagrangian}, we observe that \eqref{e:inclu1} is equivalent to \begin{align} \begin{cases} -\nabla_u \mathcal{L}_\beta(u,y) \in N_C(u)\\ A(u) = b, \end{cases} \label{e:inclu1} \end{align} which is in turn the necessary optimality condition for Program \eqref{eq:minmax}. } % %Let $\overline{u}$ be a locally optimal and suppose that there no vector $y\not=0$ such that %\begin{equation} %-\nabla L(\overline{u})^* y \in N_C(\overline{u}). %\end{equation} %Then, the first order optimality condition for $\overline{u}$ is %\begin{equation}\label{e:inclu1} %(\exists y \in\RR^m)\; - \nabla L(\overline{u})^*y - \nabla h(\overline{u}) \in N_{C}(\overline{u}). %\end{equation} %Since $\partial \iota_{\{0\}} = \RR^m$, the condition \eqref{e:inclu1} is equivalent to %\begin{equation} %0 \in \partial \iota_C(\overline{u}) +\nabla h(\overline{u}) + \nabla L(\overline{u})^* \partial \iota_{\{0\}}(L\overline{u}-b). %\end{equation} %Observe that the condition \eqref{e:inclu1} is also equivalent to the following condition %\begin{equation}\label{e:inclu2} %(\exists y \in\RR^m)\; 0 \in \nabla \mathcal{L}(\overline{u},y), %\end{equation} %where $\mathcal{L}(u,y)$ is the Lagrangian function associated to the non-linear constraint $Lu=b$, %\begin{equation} % \mathcal{L}_{\beta}(u,y) = (h +\iota_C)(u) + \scal{Lu-b}{y}. %\end{equation} %The corresponding augmented Lagrangian function associated to the non-linear constraint $Lu=b$ is defined by %\begin{equation} %(\forall \beta \in \left]0,+\infty\right[)\quad \mathcal{L}_{\beta}(u,y) = (h+\iota_C)(u) + \scal{Lu-b}{y} +\frac{1}{2\beta}\|Lu-b\|^2. %\end{equation} %For convenience, we define %\begin{equation} %(\forall \beta \in \left]0,+\infty\right[)\quad g_{\beta}(u,y) = \scal{Lu-b}{y} +\frac{1}{2\beta}\|Lu-b\|^2. %\end{equation} %and %\begin{equation} %(\forall \beta \in \left]0,+\infty\right[)\quad F_{\beta}(u,y) = h(u)+ g_{\beta}(u,y). %\end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \paragraph{\textbf{\emph{Gradient Mapping.}}} In nonconvex optimization, the relation between the gradient mapping and stationarity is well-understood \cite{Lan16,Hare2009,bolte2014proximal}, \edita{which we review here for completeness.} \begin{definition} \label{def:grad map} Given $u$ and $\gamma >0$, define the gradient mapping \begin{equation} G_{\beta,\gamma}(\cdot; y)\colon u\rightarrow \frac{u-u^+}{\gamma}, \label{eq:grad map} \end{equation} where $u^+=P_{C}(u-\gamma \nabla \mathcal{L}_ {\beta}(u,y))$. \end{definition} %\begin{definition} Given $u$ and $\gamma >0$, let $r_{\beta}(\xi,y)$ be a stochastic estimimate of $\nabla F_{\beta}(u,y)$, define %stochastic gradient mapping $SG_{\beta,\gamma}(\cdot; y)\colon u\mapsto \gamma^{-1}(u-u_+)$, where $u_+=\prox_{\gamma f}(u-\gamma r_{\beta}(\xi;y))$. %\end{definition} \edita{In particular, if we remove the constraints of Program \eqref{prob:01}, the gradient mapping reduces to $G_{\beta,\gamma}(u,y)=\nabla h(u) $. The gradient mapping is closely related to $\mathcal{L}_\beta$. The following standard result is proved in Appendix~\ref{sec:proof of smoothness}.} \begin{lemma}\label{lem:11} \edita{For fixed $y\in \RR^m$, suppose that $\nabla_u \mathcal{L}_{\beta}(\cdot, y)$ is $\lambda_\beta$ Lipschitz-continuous. For $u\in C$ and $\gamma \in (0, 1/\lambda_\beta)$, it holds that } \begin{equation} \label{e:deslem} \| G_{\beta,\gamma}(u;y)\|^{2}\leq \frac{2}{\gamma} (\mathcal{L}_\beta(u;y) - \mathcal{L}_\beta(u^+;y)), \end{equation} \edita{where \begin{align} \lambda_\beta & \le \lambda_h + \sqrt{m}\lambda_A \left(\|y\| + \frac{\|A(u)\|}{\beta} \right) + \frac{\|DA(u)\|_F^2}{\beta}, \label{eq:smoothness of Lagrangian} \end{align} where $DA(u)$ is the Jacobian of $A$ at $u$. } %where $u^+ =u^+(\mu,u) = P_{C}(u-\gamma \nabla F_{\beta}(u;y))$. %\end{enumerate} \end{lemma} %\begin{proof} For a fixed $y$, the function $u\mapsto h(u)+F_{\beta}(u;y)$ is $(\mathbb{L}_h+ \mathbb{L}_{\beta})$-Lipschitz. Hence, the results follows from %\cite[Lemma 3.2, Remark 3.2(i)]{bolte2014proximal}. %\end{proof} In \edita{practice}, the Lipschitz constant $\lambda_{\beta}$ is often hard to evaluate exactly \edita{and we might resort to the classic line search technique, reviewed below and proved in Appendix \ref{sec:proof of eval Lipsc cte} for completeness.} \edita{ -\begin{lemma} \label{lem:eval Lipsc cte} Fix $\rho \in (0,1)$ and ${\gamma}_0$. For $\gamma'>0$, let $u^+_{\gamma'} = P_C(u - \gamma' \nabla \mathcal{L}_\beta(u,y))$ and define +\begin{lemma} \label{lem:eval Lipsc cte} Fix $\theta \in (0,1)$ and ${\gamma}_0$. For $\gamma'>0$, let $u^+_{\gamma'} = P_C(u - \gamma' \nabla \mathcal{L}_\beta(u,y))$ and define \begin{equation*} \gamma := \max \left\{ -\gamma' ={\gamma}_0 \rho^i : +\gamma' ={\gamma}_0 \theta^i : \mathcal{L}_\beta (u^+_{\gamma'},y) \le \mathcal{L}_\beta (u,y) + \left\langle u^+_{\gamma'} - u , \nabla \mathcal{L}_{\beta}(u,y) \right\rangle + \frac{1}{2\gamma'} \| u^+_{\gamma'} - u\|^2 \right\}. \label{eq:defn of gamma line search} \end{equation*} Then, (\ref{e:deslem}) holds and moreover we have that \begin{align} - \gamma \ge \frac{\rho}{\lambda_\beta}. + \gamma \ge \frac{\theta}{\lambda_\beta}. \label{eq:moreover} \end{align} \end{lemma} } %Let $\delta $ and $\theta$ be in $\left]0,1\right[$ and $\overline{\gamma} > 0$. Define %\begin{alignat}{2}\label{e:non-lips} %&\gamma = \max\{\mu > 0| (\exists i \in\NN)(\mu= \overline{\gamma}\theta^i)\notag\\ %&F_{\beta}(u^+(\mu,u),y) \leq F_{\beta}(u,y) + \scal{u^+(\mu,u)-u}{\nabla F_{\beta}(u,y)} +\frac{\delta}{\mu}\|u-u^+(\mu,u)\|^2\}. %\end{alignat} %Now, let $\gamma$ be define as in \eqref{e:non-lips}. Then, since $u^+ =u^+(\mu,u) = P_{C}(u-\gamma \nabla F_{\beta}(u;y))$, %\begin{equation} %G_{\beta,\gamma}(u) - \nabla F_{\beta}(u;y)) \in N_{C}(u^+). %\end{equation} %Hence, $\scal{G_{\beta,\gamma}(u) - \nabla F_{\beta}(u;y)) }{u-u^+} \leq 0$. Using \eqref{e:non-lips}, we have %\begin{alignat}{2} %F_{\beta}(u^+,y) &\leq F_{\beta}(u,y) + \scal{u-u^+}{ -\nabla F_{\beta}(u,y)} +\frac{\delta}{\gamma}\|u-u^+\|^2\notag\\ %&= F_{\beta}(u,y) + \scal{u-u^+}{ G_{\beta,\gamma}(u) -\nabla F_{\beta}(u,y)} +\frac{\delta-1}{\gamma}\|u-u^+\|^2\notag\\ %&\leq F_{\beta}(u,y) - \frac{\delta-1}{\gamma}\|u-u^+\|^2, %\end{alignat} %which implies that %\begin{equation} %(1-\delta) \| G_{\beta,\gamma}(u;y)\|^{2}\leq \frac{1}{\gamma} (F_{\beta}(u;y) - F_{\beta}(u^+;y)). %\end{equation} %In particular, by taking $\delta =1/2$, we obtain \eqref{e:deslem}. \edita{Optimality conditions in Section \ref{sec:opt cnds} can also be expressed in terms of the gradient mapping. Indeed, it is straightforward to verify that $u^+$ is a first-order stationary point of Program \eqref{prob:01} if \begin{align} \begin{cases} G_{\beta,\gamma}(u,y) = 0\\ A(u^+) = b. \end{cases} \label{eq:opt grad map} \end{align} } % % % %\begin{lemma} %\label{l:solv} % Suppose that $\mathcal{L}_{\beta,\gamma}\in \mathbb{L}_d(\lambda_\beta)$. For $\gamma \in (0, 1/\lambda_\beta)$, $u^+=P_C(u-\gamma \nabla \mathcal{L}_\beta(u,y))$ is a first-order stationary point of Program (\ref{prob:01}) if % % $\nabla g_{\beta}(\cdot, y)$ is $\mathbb{L}_{\beta}$-Lipschitz continuous. Let $u\in C$ and %$\gamma \in ]0, 1/(\mathbb{L}_h+ \mathbb{L}_{\beta})[$. Suppose that $Lu^+=b$ and $\| G_{\beta,\gamma}(u;y)\|^{2} =0$. %Then $u^+$ is a stationary point, i.e., the first order optimality condition \eqref{e:inclu1}, for $ \overline{u} =u^+$, is satisfied. %\end{lemma} %\begin{proof} %Since $u^+ = P_{C}(u-\gamma \nabla F_{\beta}(u;y))$. Then % it follows that %\begin{equation} %\label{e:ver1} %G_{\gamma,\beta}(u;y) - \nabla F_{\beta}(u;y) \in N_C(u^+). %\end{equation} %Adding $\nabla F_{\beta}(u^+;y)$ to both sides of \eqref{e:ver1}, we obtain %\begin{equation} %\label{e:ver2} %G_{\gamma,\beta}(u;y) - \nabla F_{\beta}(u;y) +\nabla F_{\beta}(u^+;y) \in\partial f(u^+)+ \nabla F_{\beta}(u^+;y). %\end{equation} %Using the Lipschitzian gradient of $F_{\beta}(\cdot, y)$ and $\gamma (\mathbb{L}_\beta +\mathbb{L}_{h}) \leq 1$, we see that %\begin{alignat}{2} %&\|G_{\gamma,\beta}(u;y) - \nabla F_{\beta}(u;y) - \nabla F_{\beta}(u^+;y) \| \notag\\ %&\quad\leq \|G_{\gamma,\beta}(u;y)\|+\|\nabla F_{\beta}(u;y) - \nabla F_{\beta}(u^+;y) \|\notag\\ %&\quad\leq \|G_{\gamma,\beta}(u;y)\|+( \mathbb{L}_\beta +\mathbb{L}_{h})\|u-u^+\|\notag\\ %&\quad =0, %\end{alignat} %which means that $G_{\gamma,\beta}(u;y) - \nabla F_{\beta}(u;y) - \nabla F_{\beta}(u^+;y)=0$. Hence, we derive from %\eqref{e:ver2} that %\begin{equation} %\label{conkhi} %0 \in\partial f(u^+)+ \nabla F_{\beta}(u^+;y). %\end{equation} %By definition of $F_{\beta}(u^+;y)$ and $Lu^+=b$, we have %\begin{alignat}{2} %\nabla F_{\beta}(u^+;y) &= \nabla h(u^+) + \nabla L(u^+)^*y + \frac{1}{\beta} \nabla L(u^+)^*(Lu^+-b) \notag\\ %&= \nabla h(u^+) + \nabla L(u^+)^*y, %\end{alignat} %which together with \eqref{conkhi}, shows that \eqref{e:inclu1} is satisfied. %\end{proof} \paragraph{\textbf{\emph{\edita{Sufficient Optimality Conditions.}}}} Sufficient optimality conditions for Program \eqref{prob:01} are also well understood in the litterature \cite{luenberger1984linear,rockafellar2009variational,mordukhovich2006variational,gfrerer2015complete}. Indeed, $u$ is a local minimizer of Program \eqref{prob:01} if there exists $y$ for which \begin{align} \begin{cases} v^\top \left( \nabla_{uu} h(u) + \sum_{i=1}^m \nabla_{uu} A_i(u) \right) v \ge 0, \qquad \forall v \in T_C(u),\\ A(u) = b. \end{cases} \label{eq:suff cnd} \end{align} \notea{Why does above look different from sufficient cnds for Lagrangian? Suppose to be equivalent problems.} %For simple, let us assume that $C = \menge{u}{g(u)\leq c}$ for some $\mathcal{C}^2$-function $g\colon\RR^d\to\RR$. In this section we assume that $h$ and $L$ are too $\mathcal{C}^2$-functions. %Let $\overline{u}$ be a point such that $g(\overline{u})=c$ and $0\not\in\nabla g(\overline{u})$. Then, \cite[Proposition 10.3]{rockafellar2009variational} implies that %\begin{equation} %N_{C}(\overline{u}) =\menge{\mu \nabla g(\overline{u})}{\mu \geq 0}. %\end{equation} %Now suppose that the first oder optimality condition \eqref{e:inclu1} is satisfied for $\overline{u}$. Then there exists $\mu \geq 0$ and $y \in\RR^m$ such that %\begin{equation}\label{e:inclu1} % \nabla L(\overline{u})^*y + \nabla h(\overline{u}) +\mu \nabla g(\overline{u})=0. % \end{equation} % Therefore, inview of \cite[Chapter 11]{luenberger1984linear}, $\overline{u}$ is a local minimum provided that % the Hessian matrix % \begin{equation} % H(\overline{u}) = \nabla^2h(\overline{u}) + y^T\nabla^2 L(\overline{u}) + \mu \nabla^2 g(\overline{u}) % \end{equation} % is positive definite on the space %\begin{equation} %E= \menge{y}{\nabla L(\overline{u})y =0, \nabla g(\overline{u})y=0}. %\end{equation} %In our examples, this condition is checkable where $g(u) = \frac{1}{2}\|u\|^2$, $h$ is quadratic and $Lu = Muu^\top$. \section{Algorithm $\&$ Convergence} \subsection{Algorithm} We propose the following method for solving the problem \eqref{prob:01} where, the main idea is that we do a projected gradient descent step on $u$ to obtain $u^+$ and update the penalty parameter $\beta^+$ in such a way that the feasiblity $\frac{1}{2\beta^+\gamma}\|Lu^+-b\|^2$ reduce faster than the gradient mapping up to some noise level $\omega$: \begin{equation} \frac{1}{2\beta^+\gamma}\|Lu^+-b\|^2\leq \frac{1}{8} \|G_{\beta,\gamma}(u,y) \|^2 + \frac{\omega}{\gamma} \end{equation} Then update the corresponding the multiplier $y$ as in the classical ADMM: \begin{equation} y^+ = y+\frac{1}{\sigma}(LU^+-b). \end{equation} The formal algorithm is presented as follows. \begin{algorithm} \label{Algo:2} Input: $\beta_0> 0$, $c > 0$, $\alpha \in \left[0,1\right[$, $u_{0}\in \mathcal{C}$, $y_0 =0$, $\epsilon_1\in \left]0,1\right[$. Given $\beta_k$, choose $\gamma_{k} \leq \frac{1-\epsilon_1}{\mathbb{L}_{h}+\mathbb{L}_{\beta_{k}}}.$ Iterate \\ For k=0,1,\ldots\\ 1. Projected gradient: $ u_{k+1} = P_{\mathcal{C} }(u_{k} - \gamma_{k} \nabla F_{\beta_{k}}(u_{k},y_{k})).$\\ 2. Line search step\\ \quad $s=0, d_{k,0}=2$, $\beta_{k+1,0}= \frac{1}{2} \|Lu_{k+1}-b \|^2 \bigg( \frac{\gamma_{k}}{8} \|G_{\beta_{k},\gamma_{k}}(u_k,y_{k}) \|^2 + \frac{d_{k,s}}{{(1+k)}^{1+\epsilon_1}}\bigg)^{-1}.$\\ While $\beta_{k+1,s} \geq c/(k+1)^{\alpha}$ do \begin{alignat}{2}\label{e:mc} % \beta_{k+1}\geq \frac{1}{2} \|Lu_{k+1}-b \|^2 \bigg( \frac{\gamma_{k}}{8} \|G_{\beta_{k},\gamma_{k}}(u_k,y_{k}) \|^2 + \frac{d_k}{(k+1)^\alpha}\bigg)^{-1}. d_{k,s+1} &= 2*d_{k,s} \\ \beta_{k+1,s+1}&= \frac{1}{2} \|Lu_{k+1}-b \|^2 \bigg( \frac{\gamma_{k}}{8} \|G_{\beta_{k},\gamma_{k}}(u_k,y_{k}) \|^2 + \frac{d_{k,s+1}}{{(1+k)}^{1+\epsilon_1}}\bigg)^{-1}\\ s&\leftarrow s+1. \end{alignat} Endwhile\\ 3. Update $\beta_{k+1} = \beta_{k+1,s}$.\\ 4. Chose $\sigma_{k+1} \geq 2\beta_{k}$ and update $y_{k+1} = y_{k} + \frac{1}{\sigma_{k+1}} (Lu_{k+1}-b)$.\\ \end{algorithm} \begin{remark} The updating rule of $(\beta_k)_{k\in\NN}$ in \eqref{e:mc} plays a role in our analysis. Intuitively, if $u_{k+1}$ is solution then $Lu_{k+1}=b$ and \eqref{e:mc} is trivially satisfied for any $\beta_{k+1}\geq 0$. Hence $\beta_{k+1}$ enforces $u_{k+1}$ close to $\menge{u}{Lu=b}$ \end{remark} \begin{remark}When $\sigma_k\equiv \infty$, we get $y_k\equiv 0$ and hence the step 2 disappears. If we chose $\sigma_k = c(k+1)^{\alpha_1}\|Lu_k-b\|$ where $c,\alpha_1$ is chosen such that $\sigma_{k} > 2\beta_{k-1}$, then \begin{equation} \|y_{k+1}\| \leq \|y_k\| + \frac{\|Lu_{k+1}-b\|}{\sigma_{k+1}} = \|y_k\| + \frac{1}{c(k+2)^\alpha}. \end{equation} Since $\sum_{k\in\NN} \frac{1}{c(k+2)^\alpha} <+\infty$, $(\|y_k\|)_{k\in\NN}$ converges and hence bounded. Therefore, \begin{equation} b_0 = \inf_{k\in\NN} \mathcal{L}_{\beta_{k}}(u_{k+1},y_{k}) \geq \inf_{k} h(u_{k}), \end{equation} which implies that $b_0>-\infty$ whenever $(u_k)_{k\in\NN}$ or $\dom(f)$ is bounded. \end{remark} \subsection{Convergence} In view of Lemma \ref{l:solv}, we need to estimate gradient mapping $\|G_{\beta_{k},\gamma_{k}}(u_k,y_{k})\|$ as well as feasibility $\|Lu_{n+1}-b\|^2$. \begin{theorem} \label{t:1} Suppose that $b_0 = \inf_{k\in\NN} \mathcal{L}_{\beta_{k}}(u_{k+1},y_{k}) > -\infty$ and that $z_0=\sum_{k=1}^\infty \frac{d_{k,s_k}}{(1+k)^{1+\epsilon_1}} <+\infty$, where $s_k$ be the smallest index such that $\beta_{k,s_k} < c/(k+1)^{\alpha}$. Then the following hold. \begin{equation}\label{e:mapp1} \sum_{k=1}^\infty \gamma_{k} \|G_{\beta_{k},\gamma_{k}}(u_k,y_{k}) \|^2 \leq 4(\mathcal{L}_{\beta_0}(u_{1},y_0) + z_0-b_0+\frac{\gamma_0}{8}\|G_{\beta_0,\gamma_0}(u_0)\|^2), \end{equation} and \begin{equation}\label{e:feas1} \sum_{k=1}^\infty \frac{1}{\beta_{k+1}} \|Lu_{k+1}-b \|^2 \leq (\mathcal{L}_{\beta_0}(u_{1},y_0) + 3z_0-b_0+\frac{\gamma_0}{8}\|G_{\beta_0,\gamma_0}(u_0)\|^2). \end{equation} \end{theorem} % \begin{proof} Set $e_{k+1}= \frac{d_{k,s_k}}{(k+1)^\alpha}$. Then $z_0= \sum_{k\in\NN}e_k <+\infty$. % It follows from Lemma \ref{lem:11} that % \begin{alignat}{2} %G_k= \frac{\gamma_{k}}{2}\|G_{\beta_k,\gamma_k}(u_k,y_k) \|^2 &\leq \mathcal{L}_{\beta_k}(u_k,y_k) - \mathcal{L}_{\beta_{k}}(u_{k+1},y_k)\notag\\ % &= h(u_k) - h(u_{k+1}) + g_{\beta_k}(u_k,y_k) -g_{\beta_{k}}(u_{k+1},y_k)\notag\\ % &=h(u_k) - h(u_{k+1}) + g_{\beta_{k-1}}(u_k,y_{k-1}) -g_{\beta_{k}}(u_{k+1},y_k)\notag\\ % &\quad + g_{\beta_k}(u_k,y_k)- g_{\beta_{k-1}}(u_k,y_{k-1}), \label{e:sa1} % \end{alignat} % where we set $g_{\beta}(u,y) = \scal{Lu-b}{y} +\frac{1}{2\beta}\|Lu-b\|^2$. Let us estimate the last term in \eqref{e:sa1}. We have % \begin{alignat}{2} % \omega_{1,k}= g_{\beta_k}(u_k,y_k)- g_{\beta_{k-1}}(u_k,y_{k-1}) = \big(\frac{1}{2\beta_k} -\frac{1}{2\beta_{k-1}}\big)\|Lu_k-b\|^2+\scal{Lu_k-b}{y_k-y_{k-1}}. % \end{alignat} % Since $y_{k} = y_{k-1} + \frac{1}{\sigma_{k}} (Lu_{k}-b)$ and use \eqref{e:mc}, we get % \begin{alignat}{2} % &\quad \omega_{1,k} = \big(\frac{1}{2\beta_k} -\frac{1}{2\beta_{k-1}}\big)\|Lu_k-b\|^2+ \frac{1}{\sigma_k} \|Lu_k-b\|^2. % %&\leq \big(\frac{1}{2\beta_k} -\frac{1}{2\beta_{k-1}}\big)\|Lu_k-b\|^2+ \frac{\gamma_{k-1}}{8} \|G_{\beta_{k-1},\gamma_{k-1}}(u_{k-1},y_{k-1}) \|^2 + \frac{d_{k-1}}{k^\alpha}\notag\\ % %&\leq \big(\frac{1}{2\beta_k} -\frac{1}{2\beta_{k-1}}\big)\|Lu_k-b\|^2+\frac{1}{4} G_k+ \frac{1}{4} G_{k-1} + \frac{d_{k-1}}{k^\alpha}. % \end{alignat} % Let us estimate the first term in \eqref{e:sa1}. Set $T_k= h(u_k) +\scal{Lu_k-b}{y_{k-1}} $. Then % \begin{alignat}{2} % \omega_{2,k}&= h(u_k) - h(u_{k+1}) + g_{\beta_{k-1}}(u_k,y_{k-1}) -g_{\beta_{k}}(u_{k+1},y_k)\notag\\ % & = T_k -T_{k+1} + \frac{1}{2\beta_{k-1}} \|Lu_k-b\|^2 - \frac{1}{2\beta_k}\|Lu_{k+1}-b\|^2. % \end{alignat} % Therefore, we derive from \eqref{e:sa1} that % \begin{alignat}{2} %G_k &\leq \omega_{1,k} +\omega_{2,k}\notag\\ % &= T_k -T_{k+1} + \frac{1}{2\beta_k} \big(\|Lu_k-b\|^2 -\|Lu_{k+1}-b\|^2\big)+\frac{1}{\sigma_k} \|Lu_k-b\|^2.\label{e:sa2} % \end{alignat} %Now using the condition \eqref{e:mc}, we obtain %\begin{equation} % \frac{1}{2\beta_k} \|Lu_k-b\|^2 \leq \frac{1}{4} G_{k-1} + e_k\leq \frac{1}{4}G_k+ \frac{1}{4} G_{k-1} + e_k, %\end{equation} %Therefore, it follows from \eqref{e:sa2} that % \begin{alignat}{2} % \frac12 G_k &\leq (T_k+\frac{1}{4} G_{k-1}) -(T_{k+1} +\frac{1}{4} G_k )+ \frac{1}{\sigma_k} \|Lu_k-b\|^2 - \frac{1}{2\beta_k}\|Lu_{k+1}-b\|^2 + e_k\notag\\ % &\leq (T_k+\frac{1}{4} G_{k-1}) -(T_{k+1} +\frac{1}{4} G_k )+ \frac{1}{2\beta_{k-1}} \|Lu_k-b\|^2 - \frac{1}{2\beta_k}\|Lu_{k+1}-b\|^2 + e_k. \label{e:sa3} % \end{alignat} % For every $N\in\NN$, $N\geq 1$, summing \eqref{e:sa3} from $k=1$ to $k=N$, we obtain, % \begin{alignat}{2} % \sum_{k=1}^{N}\frac12 G_k \leq T_1 +\frac{1}{4} G_0 + \frac{1}{\beta_{0}} \|Lu_1-b\|^2 - T_{N+1} - \frac{1}{4}G_{N} - \frac{1}{2\beta_N} \|Lu_{N+1}-b\|^2 + z_0. % \end{alignat} % Note that, by the definiton of $T_{N+1}$, we have % \begin{alignat}{2} % -T_{N+1} - \frac{1}{2\beta_N} \|Lu_{N+1}-b\|^2& = - \mathcal{L}_{\beta_N}(u_{N+1},y_N) \leq -b_0. % \end{alignat} % Hence, % \begin{equation} % \sum_{k=1}^{N}\frac12 G_k \leq \mathcal{L}_{\beta_0}(u_{1},y_0) + z_0-b_0 +\frac{1}{4} G_0 , % \end{equation} % which proves \eqref{e:mapp1}. Moreover, \eqref{e:feas1} follows directly from \eqref{e:mc}. % \end{proof} % % \begin{corollary} Under the same condition as in Theorem \ref{t:1}. Suppose that %$\gamma_k = \mathcal{O}(\beta_k)$. Then %\begin{equation}\label{e:mapp2} %\min_{1\leq k\leq N} \|G_{\beta_{k},\gamma_{k}}(u_k,y_{k}) \|^2 = \mathcal{O}(1/N^{1-\alpha}) \to 0, %\end{equation} %and %\begin{equation}\label{e:feas2} %\min_{1\leq k\leq N} \|Lu_{k+1}-b \|^2 = \mathcal{O}(1/N) \to 0. %\end{equation} % \end{corollary} % \begin{proof} We see that % \eqref{e:mapp2} and \eqref{e:feas2} follow directly % from \eqref{e:mapp1} and \eqref{e:feas1}, respectively. % \end{proof} % \begin{corollary} \label{c:2} % Under the same condition as in Theorem \ref{t:1}. % The sequence $(F_{\beta_k}(u_k,y_{k}))_{k\in\NN}$ converges to a $F^\star$. Moreover, % if $(\|y_{k-1}\| \sqrt{\beta_k})_{k\in\NN}$ and $(\beta_{k+1}/\beta_{k})_{k\in\NN}$ are bounded by $M$, then $(h(u_k))_{k\in\NN}$ converges to $F^\star$. % \end{corollary} % \begin{proof} Note that $(F_{\beta_k}(u_{k+1},y_k))_{n\in\NN}$ is bounded below. Moreover, the proof of Theorem \ref{t:1} show that % \begin{equation} % F_{\beta_k}(u_{k+1},y_k) + \frac{\gamma_k}{8}\|\mathcal{G}_{\beta_k,\gamma_k}(u_k)\|^2 % \leq F_{\beta_{k-1}}(u_{k},y_{k-1}) + \frac{\gamma_{k-1}}{8}\|\mathcal{G}_{\beta_{k-1},\gamma_{k-1}}(u_{k-1})\|^2 +e_k. % \end{equation} % Hence $(F_{\beta_k}(u_{k+1},y_k) + \frac{\gamma_k}{8}\|\mathcal{G}_{\beta_k,\gamma_k}(u_k)\|^2)_{n\in\NN}$ converges to a finite value % $F^\star$. Since $\frac{\gamma_k}{8}\|\mathcal{G}_{\beta_k,\gamma_k}(u_k)\|^2\to 0$, we get $F_{\beta_k}(u_{k+1},y_k)\to F^\star$. Since %$\frac{1}{2\beta_k}\|Lu_{k}-b\|^2\to 0$ and $(\beta_{k+1}/\beta_{k})_{k\in\NN}$ is bounded by $M$, we obtain %\begin{equation} %\frac{1}{2\beta_k}\|Lu_{k+1}-b\|^2 =\frac{\beta_{k+1}}{2 \beta_{k+1}\beta_{k}}\|Lu_{k+1}-b\|^2 \leq \frac{M}{2\beta_{k+1}} \|L u_{k+1} -b\|^2 \to 0. %\end{equation} %Moreover, since $(\|y_{k-1}\| \sqrt{\beta_k})_{k\in\NN}$ is bounded by $M$, we also have %\begin{equation} %|\scal{Lu_k-b}{y_{k-1}}| = |\frac{1}{\sqrt{\beta_k}}\scal{Lu_k-b}{\sqrt{\beta_k}y_{k-1}}| \leq \frac{M}{\sqrt{\beta_k}}\|Lu_{k}-b\| \to 0. %\end{equation} %Therefore, %\begin{alignat}{2} %|h(u_n) - F^\star| &\leq |F_{\beta_k}(u_k,y_{k-1})-F^\star| + |\scal{Lu_k-b}{y_{k-1}}| + \frac{1}{2\beta_k}\|Lu_k-b\|^2\notag\\ %&\to 0, %\end{alignat} %which proves the desired result. % \end{proof} % \subsection{Local convergence} % Let $\overline{u}$ and $\overline{y}$ satisfy the first order optimality condition % \begin{equation} % \label{e:fr1} % -\nabla L(\overline{u})^*\overline{y} - \nabla h(\overline{u}) \in N_{C}(\overline{u}). % \end{equation} % For simple, let us recall $\nabla F_{\beta}\colon (u,y) \mapsto \nabla h(u) + \nabla L(u)^*y + \frac{1}{\beta}\nabla L(u)^*(Lu-b)$. % \begin{theorem} Under the conditions of Theorem \ref{t:1} and $\gamma_k \geq \underline{\gamma} > 0$ . % Suppose that each $\overline{u}$ in \eqref{e:fr1} is a local minima of $h$, and $C$ and $(y_k)_{k\in\NN}$ are bounded. % Then $(h(u_k))_{k\in\NN}$ converges to local optimum $h(\overline{u})$. % \end{theorem} % \begin{proof} Since $C$ is bounded, $(u_k)_{k\in\NN}$ is bounded. Therefore, there exists a subsequence % $(n_k)_{k\in\NN}$ of $\NN$ such that $u_{n_k}\to u^*$ and $y_{n_k}\to y^*$. % It follows from Theorem \ref{t:1} that $\gamma_k\|(u_{k+1}-u_k)/\gamma_k\|^2 \to 0$. Since $(\gamma_k)_{k\in\NN}$ % is bounded below by $\underline{\gamma}$, we obtain $G_{\beta_k,\gamma_k}(u_k)\to 0$. %Hence, $u_{n_k+1}\to u^*$. % Now, using the updating of $u_{k}$, we have $G_{\beta_k,\gamma_k}(u_k) -\nabla F_{\beta_k}(u_k,y_k) \in N_{C}(u_{k+1})$. % Hence, % \begin{equation}\label{e:aa2s} % (\forall u\in C)(\forall k\in\NN)\; \scal{-G_{\beta_k,\gamma_k}(u_k)+\nabla F_{\beta_{k}}(u_k,y_k) }{u-u_{k+1}}\geq 0. % \end{equation} % Since $C$ is bounded and $\nabla L$ is continuous, we obtain $\sup_{u\in C}\|\nabla L(u)\| <+\infty$, and hence % \begin{equation} % \|\frac{1}{\beta_{n_k}}\nabla L(u_{n_k})^*(Lu_{n_k}-b)\| \leq \frac{1}{c}(\sup_{u\in C}\|\nabla L(u)\| )\|Lu_{n_k}-b\|\to 0. % \end{equation} % We also have % \begin{equation} % \nabla L(u_{n_k}) \to \nabla L(u^*) \; \text{and}\; \nabla h(u_{n_k}) \to \nabla h(u^*). % \end{equation} % Since $y_{n_k}\to y^*$, we get % \begin{alignat}{2} %& \|\nabla L(u_{n_k})^*y_{n_k}- \nabla L(u^*)y^* \| \quad\\ % &\leq \|\nabla L(u_{n_k})^*y_{n_k}- \nabla L(u_{n_k})y^* \| +\|\nabla L(u_{n_k})^*y^*- \nabla L(u^*)y^* \|\notag\\ % &\leq \|\nabla L(u_{n_k})\|\|y_{n_k}- y^* \|+\|y^*\| \|\nabla L(u_{n_k})- \nabla L(u^*) \|\notag\\ % &\to 0. % \end{alignat} % Consequently, $\nabla F_{\beta_{n_k}}(u_{n_k}, y_{n_k}) \to \nabla h(u^*) +\nabla L(u^*)y^*$. Note that % $$G_{\beta_{n_k},\gamma_{n_k}}(u_{n_k})\to 0.$$ Now, passing through subsequence in \eqref{e:aa2s}, we obtain % \begin{equation}\label{e:aa2s} % (\forall u\in C)\scal{\nabla h(u^*) +\nabla L(u^*)y^* }{u-u^*}\geq 0, % \end{equation} % which is \eqref{e:fr1} for $\overline{u} = u^*$ and $\overline{y} = y^*$. By assumption, $u^*$ is local minimum and % $h(u_{n_k}) \to h(u^*)$. Therefore, $F^\star = h(u^*)$. Using Corollary \ref{c:2}, we get $h(u_k) \to h(\overline{u})$. % \end{proof} \section{Related Work \label{sec:related work}} To the best of our knowledge, the proposed method is new and different from existing methods in the literature. As mentioned in Introduction, the connection to augmented Lagrange method is already mentioned. Our method is significantly different from the augmented Lagrange method, we perform only step of the projected gradient step on primal variable $u$ instead of minimizing the augmented Lagrange fucntion. Furthermore, we update the penalty parameter $\beta$ adaptively to make sure that the feasibility reduces faster than the gradient mapping. In the case when $h=0$, a modification of Chambolle-Pock's method is investigated in \cite{Valkonen14} and preconditioned ADMM \cite{Matin17} where the convergence of iterate is proved under strong assumptions not full-filling in our setting here. %\noindent{\bf Connection to Linearized Alternating Direction Method}.\\ ADMM is the classic method proposed for solving the problem \ref{prob:01} for the case where $L$ is a linear operator and $h$ is zero \cite{gabay1976dual}. This method is an application of the Douglas-Rachford method to the dual problem \cite{Gabay83}. One of the main drawback of the ADMM is the appearance of the term $Lu$ in the update rule of $u_{k+1}$. To overcome this issue, some strategies were suggested. The first strategies is proposed in \cite{shefi2014rate}, refined in \cite{banert2016fixing}, known as alternating direction proximal method of multipliers. The second strategies is to use linearized technique \cite{lin2011linearized}. We show here that our proposed method is closed related to updating rule as the linearized alternating direction method \cite{lin2011linearized}. Assume that $h\equiv 0$ and $L$ is a linear operator. Then the proposed method can be rewritten as \begin{equation} \begin{cases} u_{k+1}= \arg\min_{u\in C} \frac{1}{2\gamma_k} \| u-u_k + \gamma_kL^*\bigg( \lambda_k + \frac{1}{\beta_k}\big(Lu_k-b\big)\bigg)\|^2\notag\\ \beta_{k+1}= \frac{1}{2} \|Lu_{k+1}-b \|^2 \bigg( \frac{\gamma_{k}}{8} \|G_{\beta_{k},\gamma_{k}}(u_k,y_{k}) \|^2 + \frac{d_k}{(k+1)^\alpha}\bigg)^{-1}\\ \text{Chose $\sigma_{k+1}\geq 2\beta_k$ and}\; \lambda_{k+1} = \lambda_k +\frac{1}{\sigma_{k+1}}(Lu_{k+1}-b), \end{cases} \end{equation} which is a variant version of Linearized ADMM \cite{lin2011linearized}. %\noindent %{\bf Connection to ALBUM3 in \cite{bolte2018nonconvex}}\\ Very recently, \cite{bolte2018nonconvex} proposes a framework with for solving the problem \ref{prob:01} with $C=\RR^d$. In particular, a special case AlBUM3 (Proximal Linearized Alternating Minimization) in this work is closely related to us where their conditions are checkable only when $L$ is linear. Moreover, our updating of $\beta_{k}$ in \cite{bolte2018nonconvex} depending on the smallest eigenvalue $L^*L$. For nonlinear $L$, the application of their method remains a challenge. %\noindent{\bf Connection to the deflected subgradient method}\\ The deflected subgradient method is investigated in \cite{burachik2010deflected} can be use to solve a special case of the Problem \ref{prob:01} for some a compact subset $\mathcal{C}$ in $\mathcal{X}$. The basis step of the deflected subgradient method to solve: given $\beta, v$, \begin{equation} u^*\in \arg\min_{u\in C} h(u) + \beta \boldsymbol{\sigma}(Lu-b) - \scal{Kv}{Lu-b} \end{equation} where $\boldsymbol{\sigma}$ is a continuous penalty function such as $\|\cdot\|$, and $K$ is bounded linear operator. In general, there is no closed -form expression for $u^*$ since it does not split $f$, $h$, $L$ invididually. Hence, it is hard to implement deflected subgradient method. This is also a common drawback of the classic penalty method and its related works \cite{gasimov2002augmented,burachik2010primal}. \section{Numerical experiments} \subsection{Hanging chain} %\begin{thebibliography}{} % % and use \bibitem to create references. Consult the Instructions % for authors for reference list style. \bibliographystyle{abbrv} \bibliography{references_alp.bib,bang.bib,ctuy16-small-bib.bib,JS_References.bib,lions.bib,references_optimal_sampling,references_yp,tythc16-small-bib,yhli.bib,bibliograpply,ctuy16-small-bib,bang1.bib,bang.bib} \appendix \edita{ \section{Proof of Lemma \ref{lem:11} \label{sec:proof of smoothness}} Note that \eqref{e:deslem} follows immediately from an application of \cite[Lemma 3.2, Remark 3.2(i)]{bolte2014proximal}. It only remains to compute the smoothness parameter of $\mathcal{L}_\beta(\cdot,y)$, namely, $\lambda_\beta$. To that end, note that \begin{align} \mathcal{L}_{\beta}(u,y) = h(u) + \sum_{i=1}^m y_i (A_i (u)-b_i) + \frac{1}{2\beta} \sum_{i=1}^m (A_i(u)-b_i)^2, \end{align} which implies that \begin{align} \nabla_u \mathcal{L}_\beta(u,y) & = \nabla h(u) + \sum_{i=1}^m y_i \nabla A_i(u) + \frac{1}{\beta} \sum_{i=1}^m (A_i(u)-b_i) \nabla A_i(u) \nonumber\\ & = \nabla h(u) + DA(u)^\top y + \frac{DA(u)^\top (A(u)-b)}{\beta}, \end{align} where $DA(u)$ is the Jacobian of $A$ at $u$. Likewise, \begin{align} \nabla^2_u \mathcal{L}_\beta(u,y) & = \nabla^2 h(u) + \sum_{i=1}^m \left( y_i + \frac{A_i(u)}{\beta} \right) \nabla^2 A_i(u) + \frac{1}{\beta} \sum_{i=1}^m \nabla A_i(u) \nabla A_i(u)^\top. \end{align} It follows that \begin{align} \|\nabla_u^2 \mathcal{L}_\beta(u,y)\| & \le \| \nabla^2 h(u) \|+ \max_i \| \nabla^2 A_i(u)\| \left (\|y\|_1+\frac{\|A(u)-b\|_1}{\beta} \right) + \frac{1}{\beta} \sum_{i=1}^m \|\nabla A_i(u)\|^2 \nonumber\\ & \le \lambda_h+ \sqrt{m} \lambda_A \left (\|y\|+\frac{\|A(u)-b\|}{\beta} \right) + \frac{\|DA(u)\|^2_F}{\beta} \qquad \left(h \in \mathbb{L}(\lambda_h),\,\, A_i \in \mathbb{L}(\lambda_A) \right) \nonumber\\ & = \lambda_h+ \sqrt{m} \lambda_A \left (\|y\|+\frac{\|A(u)-b\|}{\beta} \right) + \frac{ \|DA(u)\|_F^2}{\beta}, \end{align} and, consequently, \begin{align} \lambda_\beta & = \sup_u \|\nabla_u^2\mathcal{L}_\beta(u,y)\| \nonumber\\ & \le \lambda_h + \sqrt{m}\lambda_A \left(\|y\| + \frac{\|A(u)-b\|}{\beta} \right) + \frac{\|DA(u)\|_F^2}{\beta}, \end{align} which completes the proof of Lemma \ref{lem:11}. } \edita{ \section{Proof of Lemma \ref{lem:eval Lipsc cte} \label{sec:proof of eval Lipsc cte}} Since $u,u^+_\gamma \in C$, it holds that \begin{align} u^+_\gamma - u \in - T_C(u^+_\gamma). \label{eq:both C feas} \end{align} Also, recalling $u^+_{\gamma}$ in Definition \ref{def:grad map}, we have that \begin{equation} u^+_{\gamma} - u +\gamma \nabla \mathcal{L}_\beta(u,y) \in -N_C(u^+_{\gamma}). \label{eq:optimality of uplus} \end{equation} Lastly, $\gamma$ by definition in \eqref{eq:defn of gamma line search} satisfies \begin{align} & \mathcal{L}_{\beta}(u^+_{\gamma},y) \nonumber\\ & \le \mathcal{L}_\beta(u,y) + \left\langle u^+_{\gamma} - u , \nabla \mathcal{L}_\beta (u,y) \right\rangle + \frac{1}{2\gamma}\|u^+_{\gamma} - u\|^2 \nonumber\\ & = \mathcal{L}_\beta(u,y) + \frac{1}{\gamma} \left\langle u^+_{\gamma} - u ,u^+_\gamma - u+ \gamma \nabla \mathcal{L}_\beta (u,y) \right\rangle - \frac{1}{2\gamma}\|u^+_{\gamma} - u\|^2 \nonumber\\ & \le \mathcal{L}_\beta(u,y) - \frac{1}{2\gamma}\|u^+_{\gamma} - u\|^2 \qquad \text{(see (\ref{eq:both C feas},\ref{eq:optimality of uplus}))} \nonumber\\ & = \mathcal{L}_\beta(u,y) - \frac{\gamma}{2} \|G_{\beta,\gamma}(u,y)\|^2, \qquad \text{(see Definition \ref{def:grad map})} \end{align} which completes the proof of Lemma \ref{lem:eval Lipsc cte}. } \section{Draft of convergence proof} %We use the shorthand %\begin{align} %h_k = h(u_k), %\qquad %A_k = A(u_k), %\qquad %G_k = \| G_{\beta_k,\gamma_k}(u_k,y_k) \|^2. %\end{align} -For convenience, let us recall that +For convenience, let us recall the updates of the algorithm in iteration $k$, namely, \begin{align} u_{k+1} & = P_C( u_k - \gamma_k \nabla \mathcal{L}_{\beta_k} (u_k,y_k) ) \nonumber\\ & = P_C\left( u_k - \gamma_k \nabla h(u_k) - \gamma_k DA(u_k) ^\top \left( y_k + \frac{A(u_k)-b}{\beta_k} \right) \right), \qquad \text{(see \eqref{eq:Lagrangian})} \label{eq:update uk recall} \end{align} \begin{align} y_{k+1} =y_k + \frac{A(u_{k+1})-b}{\sigma_{k+1}}, \label{eq:y update recall} \end{align} \begin{equation} G_k = G_{\beta_k,\gamma_k}(u_k,y_k) = \frac{u_k-u_{k+1}}{\gamma_k}. \qquad \text{(see \eqref{eq:grad map})} \label{eq:grad map recall} \end{equation} -For an integer $k_0$ to be set later, suppose that $k\ge k_0$. -In order to apply Lemma \ref{lem:11} in every iteration, we set $\gamma_k$ to be the output of the line search subroutine in Lemma \ref{lem:eval Lipsc cte}. -%assume that -%\begin{align} -%\gamma_k \le \ol{\gamma}_k, -%\qquad \forall k \ge k_0, -%\label{eq:smoothness emp} -%\end{align} -%We can make the assumption in \eqref{eq:smoothness emp} more concrete as follows. As a consequence of (\ref{eq:smoothness of Lagrangian},\ref{eq:moreover}), note that -%\begin{align} -%\frac{\rho}{\ol{\gamma}_k} -%& \le -% \lambda_h + \sqrt{m}\lambda_A \left(\|y_{k}\|+\frac{\|A(u_k)-b\|}{\beta_k} \right)+ \frac{\|DA(u_k)\|_F^2}{\beta_k} -%\qquad \text{(see (\ref{eq:smoothness of Lagrangian},\ref{eq:moreover}))} \nonumber\\ -%& -%\le \lambda_h + \sqrt{m}\lambda_A \left( \|y_{k}\| +\frac{\|A(u_k)-b\|}{\beta_k} \right)+ \frac{m \|DA(u_k)\|^2}{\beta_k} \nonumber\\ -%& \le \lambda_h + \sqrt{m}\lambda_A \left( \|y_{k}\| +\frac{\|A(u_k)-b\|}{\beta_k} \right)+ \frac{m \eta_{\max}^2}{\beta_k}, -%\qquad \text{(see \eqref{eq:defn etaMax})} -%\end{align} -%where $\rho$ is the parameter of the line search subroutine and we set -%\begin{align} -%\eta_{\max} := \max_{u\in C} \|DA(u)\|. -%\label{eq:defn etaMax} -%\end{align} -%Therefore, \eqref{eq:smoothness emp} follows from the assumptions -%\begin{align} -%\gamma_k \le \frac{\rho}{4\lambda_h} \wedge -%\frac{\rho}{4m\lambda_A \|y_{k}\|} -%\wedge -%\frac{\rho \beta_k}{4\|A(u_k)-b\|} -%\wedge -%\frac{\rho \beta_k}{4m\eta_{\max}^2}, -%\qquad \forall k\ge k_0. -%\label{eq:smoothness emp suff} -%\end{align} -%where $\wedge$ stands for minimum. -We may now apply Lemma \ref{lem:11} to find that +For integers $k_0\le k_1$, consider the interval +\begin{equation} + K = [k_0:k_1]=\{k_0,\cdots,k_1\}. + \end{equation} +Since $\gamma_k$ is determined by the line search subroutine in Lemma \ref{lem:eval Lipsc cte}, we may now apply Lemma \ref{lem:11} for every iteration in this interval to find that \begin{align} \frac{\gamma_k \|G_k\|^2}{2} & \le \mathcal{L}_{\beta_k}(u_k,y_k) - \mathcal{L}_{\beta_k}(u_{k+1},y_k) \qquad \text{(see Lemma \ref{lem:11})} \nonumber\\ -& = h(u_k) - h(u_{k+1})+ \langle A(u_k)-A(u_{k+1}) , y_k \rangle+ \frac{\|A(u_k)-b\|^2 - \|A(u_{k+1})-b\|^2}{2\beta_k}. +& = h(u_k) - h(u_{k+1})+ \langle A(u_k)-A(u_{k+1}) , y_k \rangle+ \frac{\|A(u_k)-b\|^2 - \|A(u_{k+1})-b\|^2}{2\beta_k}, \qquad \text{(see \eqref{eq:Lagrangian})} \label{eq:smoothness rep} \end{align} -We now note that +for every $k\in K$. +On the other hand, \begin{align} y_k = y_{k_0} + \sum_{i=k_0+1}^k \frac{A(u_i)-b}{\sigma_i}, \qquad \text{(see \eqref{eq:y update recall})} +\label{eq:y update unfolded} \end{align} -which after substituting in \eqref{eq:smoothness rep} yields that +which, after substituting in \eqref{eq:smoothness rep}, yields that \begin{align} \frac{\gamma_k \|G_k\|^2}{2} & \le h(u_k) - h(u_{k+1}) + \left\langle A(u_k) - A(u_{k+1}) , y_{k_0} + \sum_{i=k_0+1}^k \frac{A(u_i)-b}{\sigma_i} \right\rangle +\frac{\|A(u_k)-b\|^2-\|A(u_{k+1})-b\|^2}{2\beta_k}. \label{eq:key ineq} \end{align} -Additionally, let us take +Additionally, let us assume that \begin{align} -\beta_k \le \sigma_k, -\qquad \forall k \ge k_0. +\beta_k = \frac{\beta}{\sqrt{k}}, +\qquad +\sigma_k = \beta k, +\qquad \forall k \in K, \label{eq:beta n sigma} \end{align} -By summing up the key inequality in \eqref{eq:key ineq} over $k$ from $k_0$ to $K$ and using \eqref{eq:beta n sigma}, we find that +with $\beta>0$. For the record, the above assumptions imply that \begin{align} -\sum_{k=k_0}^K \frac{\gamma_k \|G_k\|^2}{2} -& \le h(u_{k_0}) - h(u_{K+1}) + \langle A(u_{k_0}) - A(u_{K+1}) , y_{k_0}\rangle -+ \sum_{k=k_0}^K \sum_{i=k_0+1}^k \left\langle A(u_k) - A(u_{k+1}) , \frac{A(u_i)-b}{\sigma_i} \right\rangle \nonumber\\ -& \qquad + \sum_{k=k_0}^K \frac{\|A(u_k)-b\|^2}{2\beta_k} - \sum_{k=k_0}^K \frac{\|A(u_{k+1})-b\|^2}{2\beta_k} \nonumber\\ -& = h(u_{k_0}) - h(u_{K+1}) + \langle A(u_{k_0}) - A(u_{K+1}) , y_{k_0} \rangle -+ \sum_{k=k_0}^K \sum_{i=k_0+1}^k \left\langle A(u_k) - A(u_{k+1}) , \frac{A(u_i)-b}{\sigma_i} \right\rangle\nonumber\\ -& \qquad + \sum_{k=k_0}^K \frac{\|A(u_k)-b\|^2}{2\beta_k} - \sum_{k=k_0+1}^{K+1} \frac{\|A(u_{k})-b\|^2}{2\beta_{k-1}} \nonumber\\ -& = h(u_{k_0}) - h(u_{K+1}) + \langle A(u_{k_0}) - A(u_{K+1}) , y_{k_0} \rangle + \frac{\|A(u_{k_0})-b\|^2}{2\beta_{k_0}} + \sum_{i=k_0+1}^K \sum_{k=i}^K \left\langle A(u_k) - A(u_{k+1})) , \frac{A(u_i)-b}{\sigma_i} \right\rangle +\frac{1}{\beta_k}- \frac{1}{\beta_{k-1}} \le \frac{1 }{\beta \sqrt{k}}, +\qquad +\frac{1}{\sigma_k} \le \frac{1}{\beta \sqrt{k}}, +\qquad \forall k\in K, +\label{eq:consequences} +\end{align} +for sufficiently large $k_0$. +By summing up the key inequality in \eqref{eq:key ineq} over $k$ from $k_0$ to $k_1$ and using \eqref{eq:beta n sigma}, we find that +\begin{align} +& \sum_{k=k_0}^{k_1} \frac{\gamma_k \|G_k\|^2}{2} \nonumber\\ +& \le h(u_{k_0}) - h(u_{k_1+1}) + \langle A(u_{k_0}) - A(u_{k_1+1}) , y_{k_0}\rangle ++ \sum_{k=k_0}^{k_1} \sum_{i=k_0+1}^k \left\langle A(u_k) - A(u_{k+1}) , \frac{A(u_i)-b}{\sigma_i} \right\rangle + \nonumber\\ +& \qquad + \sum_{k=k_0}^{k_1} \frac{\|A(u_k)-b\|^2}{2\beta_k} - \sum_{k=k_0}^{k_1} \frac{\|A(u_{k+1})-b\|^2}{2\beta_k} +\qquad \text{(see \eqref{eq:key ineq})} +\nonumber\\ +& = h(u_{k_0}) - h(u_{{k_1}+1}) + \langle A(u_{k_0}) - A(u_{{k_1}+1}) , y_{k_0} \rangle ++ \sum_{k=k_0}^{k_1} \sum_{i=k_0+1}^k \left\langle A(u_k) - A(u_{k+1}) , \frac{A(u_i)-b}{\sigma_i} \right\rangle\nonumber\\ +& \qquad + \sum_{k=k_0}^{k_1} \frac{\|A(u_k)-b\|^2}{2\beta_k} - \sum_{k=k_0+1}^{{k_1}+1} \frac{\|A(u_{k})-b\|^2}{2\beta_{k-1}} \nonumber\\ +& \le h(u_{k_0}) - h(u_{{k_1}+1}) + \langle A(u_{k_0}) - A(u_{{k_1}+1}) , y_{k_0} \rangle + \frac{\|A(u_{k_0})-b\|^2}{2\beta_{k_0}} + \sum_{i=k_0+1}^{k_1} \sum_{k=i}^{k_1} \left\langle A(u_k) - A(u_{k+1}) , \frac{A(u_i)-b}{\sigma_i} \right\rangle \nonumber\\ & \qquad -+ \sum_{k=k_0+1}^K \left( \frac{1}{2\beta_k} - \frac{1}{2\beta_{k-1}} \right) \|A(u_k)-b\|^2 -- \frac{\|A(u_{K+1})-b\|^2}{2\beta_K} ++ \sum_{k=k_0+1}^{k_1} \left( \frac{1}{2\beta_k} - \frac{1}{2\beta_{k-1}} \right) \|A(u_k)-b\|^2 \nonumber\\ -& =: \mu + \sum_{i=k_0+1}^K \left\langle A(u_i) - A(u_{K+1}), \frac{A(u_i)-b}{\sigma_i} \right\rangle + \sum_{k=k_0+1}^K \left( \frac{1}{2\beta_k} - \frac{1}{2\beta_{k-1}} \right) \|A(u_k)-b\|^2 -- \frac{\|A(u_{K+1})-b\|^2}{2\beta_K} +& \le \mu + \sum_{i=k_0+1}^{k_1} \left\langle A(u_i) - A(u_{{k_1}+1}), \frac{A(u_i)-b}{\sigma_i} \right\rangle + \sum_{k=k_0+1}^{k_1} \left( \frac{1}{2\beta_k} - \frac{1}{2\beta_{k-1}} \right) \|A(u_k)-b\|^2 +\qquad \text{(see \eqref{eq:defn mu})} \nonumber\\ -& = \mu + \sum_{k=k_0+1}^K \left( \frac{1}{\sigma_k} +\frac{1}{2\beta_k} - \frac{1}{2\beta_{k-1}} \right) \|A(u_k)-b\|^2 -- \sum_{k=k_0+1}^K \left \langle A(u_{K+1})-b, \frac{A(u_k)-b}{\sigma_k} \right\rangle -- \frac{\|A(u_{K+1})-b\|^2}{2\beta_K} +& = \mu + \sum_{k=k_0+1}^{k_1} \left( \frac{1}{\sigma_k} +\frac{1}{2\beta_k} - \frac{1}{2\beta_{k-1}} \right) \|A(u_k)-b\|^2 +- \sum_{k=k_0+1}^{k_1} \left \langle A(u_{{k_1}+1})-b, \frac{A(u_k)-b}{\sigma_k} \right\rangle \nonumber\\ -& \le\mu + \sum_{k=k_0+1}^K \left( \frac{1}{\sigma_k}+ \frac{1}{2\beta_k}- \frac{1}{2\beta_{k-1}} \right) \|A(u_k)-b\|^2 +& \le\mu + \sum_{k=k_0+1}^{k_1} \left( \frac{1}{\sigma_k}+ \frac{1}{2\beta_k}- \frac{1}{2\beta_{k-1}} \right) \|A(u_k)-b\|^2 + - \sum_{k=k_0+1}^K \frac{\|A(u_{K+1})-b\| \|A(u_k)-b\|}{\sigma_k} - - \frac{\|A(u_{K+1})-b\|^2}{2\beta_K} \nonumber\\ -& \le \mu + \sum_{k=k_0+1}^K \left( \frac{1}{\sigma_k}+ \frac{1}{2\beta_k} \right) \|A(u_k)-b\|^2 + \sum_{k=k_0+1}^{k_1} \frac{1}{\sigma_k} \|A(u_{{k_1}+1})-b\| \|A(u_k)-b\| \nonumber\\ +& \le \mu + \sum_{k=k_0+1}^{k_1} \left( \frac{1 }{\beta \sqrt{k}}+ \frac{1}{2\beta_k}-\frac{1}{2\beta_{k-1}} \right) \|A(u_k)-b\|^2 + - \sum_{k=k_0+1}^K \frac{\|A(u_{K+1})-b\| \|A(u_k)-b\|}{\sigma_k} + \sum_{k=k_0+1}^{k_1} \frac{ 1}{\beta \sqrt{k}} \|A(u_{{k_1}+1})-b\| \|A(u_k)-b\| + \qquad \text{(see (\ref{eq:consequences}))} \nonumber\\ -& \le \mu + \sum_{k=k_0+1}^K \frac{3\|A(u_k)-b\|^2}{2\beta_k} -+ - \sum_{k=k_0+1}^K \frac{\|A(u_{K+1})-b\| \|A(u_k)-b\|}{\beta_k} -\qquad \text{(see \eqref{eq:beta n sigma})} + & \le + \mu + \sum_{k=k_0+1}^{k_1} \frac{3 }{2\beta \sqrt{k}} \|A(u_k)-b\|^2 + \sum_{k=k_0+1}^{k_1} \frac{1}{\beta \sqrt{k}} \|A(u_{{k_1}+1})-b\| \|A(u_k)-b\| + \qquad \text{(see (\ref{eq:consequences}))} \nonumber\\ - & \le \mu + \sum_{k=k_0+1}^K \frac{\sqrt{k}+3}{2\beta_k} \|A(u_k)-b\|^2 -+ \sum_{k=k_0+1}^K \frac{\|A(u_{K+1})-b\|^2}{2\sqrt{k}\beta_k}, -\label{eq:long chain} + & \le \mu + \sum_{k=k_0+1}^{k_1} \left( \frac{3}{2\beta \sqrt{k}} + \frac{1}{2\beta } \right) \|A(u_k)-b\|^2 + \sum_{k=k_0+1}^{k_1} \frac{1}{2\beta k} \|A(u_{{k_1}+1})-b\|^2 + \qquad \left( 2ab \le ca^2 + c^{-1}b^2 \right) + \nonumber\\ + & \le \mu + \sum_{k=k_0+1}^{k_1} \frac{2 }{\beta } \|A(u_k)-b\|^2 + \sum_{k=k_0+1}^{k_1} \frac{1 }{2\beta k} \|A(u_{{k_1}+1})-b\|^2 \nonumber\\ +& +\le \mu + \sum_{k=k_0+1}^{k_1} \frac{2 }{\beta} \|A(u_k)-b\|^2 + \frac{\log {k_1}}{\beta} \|A(u_{{k_1}+1})-b\|^2 +\qquad +\left( \sum_{k=1}^{k_1} \frac{1}{k} \le 2\int_{1}^{k_1} \frac{d\kappa }{\kappa} =2 \log {k_1} \right) +\nonumber\\ +& \le \mu + \frac{2 }{\beta} \sum_{k=k_0+1}^{{k_1}+1} \log k \|A(u_k)-b\|^2, + \label{eq:long chain} \end{align} -where, in the last line above, we used the inequality $2ab\le ca^2+ c^{-1}b^2$ for scalars $a,b$ and $c> 0$. In \eqref{eq:long chain}, we also assumed that +where we assumed that \begin{equation} \mu := \sup_k h(u_{k_0}) - h(u_k) + \langle A(u_{k_0})-A(u_k) ,y_{k_0}\rangle + \frac{\|A(u_{k_0})-b\|^2}{2\beta_{k_0}}< \infty. \label{eq:defn mu} \end{equation} -Let us also assume that $\{\beta_k\}_k$ is non-increasing, namely, -\begin{equation} -\beta_{k+1} \le \beta_k , -\qquad -\forall k \ge k_0, -\label{eq:sigma dec} -\end{equation} -which allows us to further simplify the last line in \eqref{eq:long chain} as -\begin{align} -\sum_{k=k_0}^K \frac{\gamma_k\|G_k\|^2}{2} -& \le -\mu + \sum_{k=k_0+1}^K \frac{\sqrt{k}+3}{2\beta_k} - \|A(u_k)-b\|^2 + - \sum_{k=k_0+1}^K \frac{\|A(u_{K+1})-b\|^2}{2\sqrt{k}\beta_k} -\nonumber\\ -& \le \mu + \sum_{k=k_0+1}^K \frac{\sqrt{k}+3}{2\beta_k} \|A(u_k)-b\|^2 + - \sum_{k=k_0+1}^K \frac{\|A(u_{K+1})-b\|^2}{2\sqrt{k}\beta_{K+1}} -\qquad \text{(see \eqref{eq:sigma dec})} -\nonumber\\ -& \le \mu + \sum_{k=k_0+1}^K \frac{\sqrt{k}+3}{2\beta_k} \|A(u_k)-b\|^2 + - \frac{\sqrt{K+1}}{\beta_{K+1}} \|A(u_{K+1})-b\|^2 - \qquad \left( \sum_{k=1}^K \frac{1}{\sqrt{k}} \le \int_{0}^K \frac{d\kappa}{\sqrt{\kappa}} = 2\sqrt{K} \right) \nonumber\\ -& \le \mu + \sum_{k=k_0+1}^{K} \frac{2\sqrt{k}}{\beta_k} \|A(u_k)\|^2 + \frac{\sqrt{K+1}}{\beta_{K+1}} \|A(u_{K+1})-b\|^2 \nonumber\\ -& \le \mu + \sum_{k=k_0+1}^{K+1} \frac{2\sqrt{k}}{\beta_k} \|A(u_k)-b\|^2. -\label{eq:raw} -\end{align} +% +%Let us also assume that $\{\beta_k\}_k$ is non-increasing, namely, +%\begin{equation} +%\beta_{k+1} \le \beta_k , +%\qquad +%\forall k \ge k_0, +%\label{eq:sigma dec} +%\end{equation} +%which allows us to further simplify the last line in \eqref{eq:long chain} as +%\begin{align} +%\sum_{k=k_0}^K \frac{\gamma_k\|G_k\|^2}{2} +%& \le +%\mu + \sum_{k=k_0+1}^K \frac{\sqrt{k}+3}{2\beta_k} +% \|A(u_k)-b\|^2 + +% \sum_{k=k_0+1}^K \frac{\|A(u_{K+1})-b\|^2}{2\sqrt{k}\beta_k} +%\nonumber\\ +%& \le \mu + \sum_{k=k_0+1}^K \frac{\sqrt{k}+3}{2\beta_k} \|A(u_k)-b\|^2 + +% \sum_{k=k_0+1}^K \frac{\|A(u_{K+1})-b\|^2}{2\sqrt{k}\beta_{K+1}} +%\qquad \text{(see \eqref{eq:sigma dec})} +%\nonumber\\ +%& \le \mu + \sum_{k=k_0+1}^K \frac{\sqrt{k}+3}{2\beta_k} \|A(u_k)-b\|^2 + +% \frac{\sqrt{K+1}}{\beta_{K+1}} \|A(u_{K+1})-b\|^2 +% \qquad \left( \sum_{k=1}^K \frac{1}{\sqrt{k}} \le \int_{0}^K \frac{d\kappa}{\sqrt{\kappa}} = 2\sqrt{K} \right) \nonumber\\ +%& \le \mu + \sum_{k=k_0+1}^{K} \frac{2\sqrt{k}}{\beta_k} \|A(u_k)\|^2 + \frac{\sqrt{K+1}}{\beta_{K+1}} \|A(u_{K+1})-b\|^2 \nonumber\\ +%& \le \mu + \sum_{k=k_0+1}^{K+1} \frac{2\sqrt{k}}{\beta_k} \|A(u_k)-b\|^2. +%\label{eq:raw} +%\end{align} %We now try to formalize Volkan's intuition that gradient mapping should derive down the feasibility gap. Let us assume that %\begin{align} %\sum_{k=1}^{K+1} \frac{\|A_k\|^2}{\beta_k} \le \sum_{k=0}^K \gamma_k G_k^2. %\label{eq:assumption} %\end{align} %And we take %\begin{equation} %\sigma_k = 6 \beta_k, %\qquad \forall k. %\end{equation} %Then, by combining the two inequalities above, we reach %\begin{align} % \sum_{k=1}^{K+1} \frac{\|A_k\|^2}{\beta_k} \le 4\mu_0, %\end{align} %\begin{align} %\sum_{k=0}^K \gamma_k G_k^2 \le 4\mu_0. %\end{align} %That is, Volkan's assumption \eqref{eq:assumption} successfully bounds both the gradient mapping and the feasibility gap. One question is the interplay between $\{\beta_k,\gamma_k\}_k$ to ensure the validity of Volkan's assumption, which feels like some sort of \emph{uncertainty principle}. -Note that \eqref{eq:raw} bounds the gradient mapping with the feasibility gap. We next find a converse, thus bounding the feasibility gap with the gradient mapping. To that end, let $T_C(u)$ and $P_{T_C(u)}$ be the tangent cone of $C$ at $u\in C$ and orthogonal projection onto this subspace, respectively. Likewise, let $N_C(u)$ and $P_{N_{C}(u)}$ be the normal cone of $C$ at $u$ and the corresponding orthogonal projection. +Note that \eqref{eq:long chain} bounds the gradient mapping with the feasibility gap. We next find a converse, thus bounding the feasibility gap with the gradient mapping. To that end, let $T_C(u)$ and $P_{T_C(u)}$ be the tangent cone of $C$ at $u\in C$ and orthogonal projection onto this subspace, respectively. Likewise, let $N_C(u)$ and $P_{N_{C}(u)}$ be the normal cone of $C$ at $u$ and the corresponding orthogonal projection. The update rule for $u_k$ in \eqref{eq:update uk recall} immediately implies that \begin{align} G_k - \nabla h(u_k) - DA(u_k) ^\top y_k- \frac{1}{\beta_k} DA(u_k)^\top (A(u_k)-b) \in N_C(u_{k+1}). \label{eq:opt cnd of update} \end{align} By definition in \eqref{eq:grad map recall}, we have that $G_k \in T_C(u_{k+1})$ which, together with \eqref{eq:opt cnd of update}, imply that \begin{align} G_k & = P_{T_C(u_{k+1})} \left( - \nabla h(u_k) - DA(u_k) ^\top y_k- \frac{1}{\beta_k} DA(u_k)^\top (A(u_k)-b) \right) \nonumber\\ & = P_{T_C(u_{k+1})}(- \nabla h(u_k)) + P_{T_C(u_{k+1})}(- DA(u_k) ^\top y_k) + \frac{1}{\beta_k} P_{T_C(u_{k+1})} ( - DA(u_k)^\top (A(u_k)-b) ) \nonumber\\ & = P_{T_C(u_{k+1})}(- \nabla h(u_k)) + P_{T_C(u_{k+1})}(- DA(u_k) ^\top y_{k-1}) + \left(\frac{1}{\beta_k}+ \frac{1}{\sigma_k} \right) P_{T_C(u_{k+1})} ( - DA(u_k)^\top (A(u_k)-b) ), \end{align} where the last line above uses \eqref{eq:y update recall}. After rearranging and applying the triangle inequality above, we reach \begin{align} \frac{1}{\beta_k}\| P_{T_C(u_{k+1})}(DA(u_k)^\top (A(u_k)-b)) \| & \le \left( \frac{1}{\sigma_k}+ \frac{1}{\beta_k} \right) \| P_{T_C(u_{k+1})} (DA(u_k)^\top (A(u_k)-b) ) \| \nonumber\\ & \le \|\nabla h(u_k)\| + \|DA(u_k) \| \cdot \|y_{k-1}\|+ \|G_k\| \nonumber\\ & \le \lambda '_h + \eta_{\max} \|y_{k-1}\|+ \|G_k\|, -\qquad \text{(see \eqref{eq:defn etaMax})} \label{eq:bnd on Ak raw} \end{align} where we set \begin{align} \lambda'_h := \max_{u\in C} \| \nabla h(u)\|, \qquad \eta_{\max}= \max_{u\in C} \|DA(u)\|. \label{eq:defn lambdap n etaMax} \end{align} We next translate \eqref{eq:bnd on Ak raw} into an upper bound on $\|A(u_k)-b\|$. \begin{lemma}\label{lem:bnd bnd Ak} %Suppose that $C$ is sufficiently smooth, in the sense that there exists a constant $\tau_C$ such that %\begin{equation} %\|P_{T_C(u)} - P_{T_C(u')}\| \le \tau_C \|u-u'\|, %\qquad \forall u,u'\in C. %\label{eq:curvature} %\end{equation} For an integer $k_0$, let \begin{align} -S_{k_0} \supseteq \bigcup_{k\ge k_0} T_{C}(u_k), +S_{K} \supseteq \bigcup_{k\in K} T_{C}(u_k), \end{align} -and, with some abuse of notation, let $S_{k_0}$ also denote an orthonormal basis for this subspace. For $\eta_{\min},\rho>0$, let us assume that +and, with some abuse of notation, let $S_{K}$ also denote an orthonormal basis for this subspace. For $\rho>0$, suppose that there exists $\eta_{\min}$ such that \begin{align} -0 < \eta_{\min} := +0 < \eta_{\min } := \begin{cases} -\min_u \, \left( S_{k_0}^\top P_{T_C(u)} (DA(u)^\top v ) \right\| \\ +\min_u \, \left\| S_{K}^\top P_{T_C(u)} (DA(u)^\top v ) \right\| \\ \|v\| =1\\ \|A(u)-b\|\le \rho\\ u\in C. \end{cases} \label{eq:new slater} \end{align} Suppose also that \begin{align} -\sup_{k\ge k_0 }\|A(u_k)-b\| \le \rho, +\sup_{k\in K }\|A(u_k)-b\| \le \rho, \label{eq:good neighb} \end{align} \begin{align} \operatorname{diam}(C) \le \frac{\eta_{\min}}{2\lambda_A}. \label{eq:cnd for well cnd} \end{align} %\begin{align} % \sup_{k\ge k_0} \gamma_k \|G_k\| \le \frac{\eta_{\min}}{2\lambda_A}. %\label{eq:cnd for well cnd} %\end{align} %\begin{align} %\sup_{k\ge k_0 }\frac{2\sigma_k}{\eta_{\min}} \left( \lambda'_h + \eta_{\max} \|y_{k-1}\|+ \|G_k\| \right) \le \rho. %\label{eq:to be met asympt} %\end{align} Then it holds that \begin{align} \|A(u_k) -b\| \le \frac{2\beta_k}{\eta_{\min}} \left( \lambda'_h + \eta_{\max} \|y_{k-1}\|+ \|G_k\| \right) , -\qquad \forall k\ge k_0. +\qquad \forall k\in K. \label{eq:bnd on Ak final} \end{align} \end{lemma} -Roughly speaking, \eqref{eq:bnd on Ak final} states that the feasibility gap is itself bounded by the gradient map. In order to apply Lemma \ref{lem:bnd bnd Ak}, let us assume that there exists $k_0$ such that (\ref{eq:good neighb},\ref{eq:cnd for well cnd}) hold. Therefore, Lemma \ref{lem:bnd bnd Ak} is in force with such $k_0$. After taking $K\ge k_0$, we may now substitute \eqref{eq:bnd on Ak final} back into \eqref{eq:raw} to find that +Roughly speaking, \eqref{eq:bnd on Ak final} states that the feasibility gap is itself bounded by the gradient map. In order to apply Lemma \ref{lem:bnd bnd Ak}, let us assume that (\ref{eq:good neighb}) holds. Lemma \ref{lem:bnd bnd Ak} is then in force and we may now substitute \eqref{eq:bnd on Ak final} back into \eqref{eq:long chain} to find that \begin{align} -\sum_{k=k_0}^K \gamma_k \|G_k\|^2 -& \le \sum_{k=k_0+1}^{K+1} \frac{4\sqrt{k}\|A(u_k)-b\|^2}{\beta_k} +2 \mu -\qquad \text{(see \eqref{eq:raw})} +\sum_{k=k_0}^{k_1} \gamma_k \|G_k\|^2 +& \le \frac{4 }{\beta} \sum_{k=k_0+1}^{{k_1}+1} \log k \|A(u_k)-b\|^2 +2 \mu +\qquad \text{(see \eqref{eq:long chain})} \nonumber\\ -& \le \sum_{k=k_0+1}^{K+1} \frac{4 \sqrt{k}\beta_k}{\eta_{\min}^2 } \left( \lambda_h' + \eta_{\max} \|y_{k-1}\| + \|G_k\| \right)^2+ 2\mu +& \le \frac{16 }{\beta \eta_{\min}^2} \sum_{k=k_0+1}^{{k_1}+1} + \beta_k^2 \log k \left( \lambda_h' + \eta_{\max} \|y_{k-1}\| + \|G_k\| \right)^2+ 2\mu \qquad \text{(see \eqref{eq:bnd on Ak final})} \nonumber\\ -& \le \sum_{k=k_0+1}^{K+1} \frac{12\sqrt{k}\beta_k}{\eta_{\min}^2 } \left( \lambda_h'^2 + \eta_{\max}^2 \|y_{k-1}\|^2 + \|G_k\|^2 \right)+ 2\mu, + & \le + \frac{16\beta}{ \eta_{\min}^2} \sum_{k=k_0+1}^{{k_1}+1} + \frac{\log k}{k} \left( \lambda_h' + \eta_{\max} \|y_{k-1}\| + \|G_k\| \right)^2+ 2\mu +\qquad \text{(see (\ref{eq:beta n sigma}))} \nonumber\\ +& \le +\frac{48\beta}{ \eta_{\min}^2} \sum_{k=k_0+1}^{{k_1}+1} + \frac{\log k}{k} \left( \lambda_h'^2 + \eta_{\max}^2 \|y_{k-1}\|^2 + \|G_k\|^2 \right)+ 2\mu. +\qquad \left( (a+b+c)^2 \le 3(a^2+b^2+c^2) \right) \label{eq:to be used for feas} \end{align} -where the last line used the inequality $(a+b+c)^2 \le 3(a^2+b^2+c^2)$ for scalars $a,b,c$. %To move forward, let us assume that %\begin{align} %\sigma_k = o(\sqrt{k}), %\qquad %\frac{\sqrt{k}\sigma_k}{\gamma_k} = o(1), %\qquad %k^{\frac{3}{2}} \sigma_k \|y_{k-1}\|^2 =o(1). %\label{eq:cnd on sigma gamma} %\end{align} -Let us assume that +To simplify the above expression, let us assume that \begin{equation} -\frac{12\sqrt{k}\beta_k}{\eta_{\min}^2} \le \frac{\gamma_k}{2}, -\qquad \forall k\ge k_0, +\frac{48\beta\log k}{ \eta_{\min}^2 k} \le \frac{\gamma_k}{2}, +\qquad \forall k\in K. \label{eq:to be used later on} \end{equation} -which we will use below. After rearranging \eqref{eq:to be used for feas} and applying \eqref{eq:to be used later on}, we arrive at +Let $|K|=k_1-k_0+1$ be the size of the interval $K$. +After rearranging \eqref{eq:to be used for feas} and applying \eqref{eq:to be used later on}, we arrive at \begin{align} -& \frac{K-k_0+1}{2} \cdot \min_{k_0 \le k\le K} \gamma_k \|G_k\|^2 \nonumber\\ -& \le \sum_{k=k_0}^K \frac{\gamma_k}{2} \|G_k\|^2 \nonumber\\ -& \le \sum_{k=k_0}^K \left(\gamma_k - \frac{12\sqrt{k}\beta_k}{\eta^2_{\min}} \right)\|G_k\|^2 +& \frac{|K|}{2} \cdot \min_{k \in K} \gamma_k \|G_k\|^2 \nonumber\\ +& \le \sum_{k=k_0}^{k_1} \frac{\gamma_k}{2} \|G_k\|^2 \nonumber\\ +& \le \sum_{k=k_0}^{k_1} \left(\gamma_k - \frac{48\beta \log k}{\eta_{\min}^2 k} \right)\|G_k\|^2 \qquad \text{(see \eqref{eq:to be used later on})} \nonumber\\ -& \le \frac{12 \lambda_h'^2}{\eta_{\min}^2} \sum_{k=k_0+1}^{K+1} \sqrt{k}\beta_k -+\frac{12\eta_{\max}^2}{\eta_{\min}^2} \sum_{k=k_0+1}^{K+1} \sqrt{k}\beta_k \|y_{k-1}\|^2 - +2\mu + & \le \frac{48 \beta \lambda'^2_h }{\eta_{\min}^2 } \sum_{k=k_0+1}^{{k_1}+1} \frac{\log k}{k} ++ \frac{48\beta \eta_{\max}^2}{\eta_{\min}^2} \sum_{k=k_0+1}^{{k_1}+1} \frac{\|y_{k-1}\|^2\log k}{k} + +\frac{48\beta \log(k_1+1) \|G_{k_1+1}\|^2}{\eta_{\min}^2 (k_1+1)} ++2\mu \qquad \text{(see \eqref{eq:to be used for feas})} \nonumber\\ -& =: \frac{12\lambda'^2_h (K-k_0+1)}{\eta_{\min}^2} M_{K} -+ \frac{12\eta_{\max}^2 (K-k_0+1)}{\eta_{\min}^2} M_{K}' -+ 2\mu, + & =: \frac{48 \beta \lambda'^2_h }{\eta_{\min}^2 } \sum_{k=k_0+1}^{{k_1}+1} \frac{\log k}{k} ++ \frac{48\beta \eta_{\max}^2}{\eta_{\min}^2} \sum_{k=k_0+1}^{{k_1}+1} \frac{\|y_{k-1}\|^2\log k}{k} ++2\mu' \nonumber\\ + & \le \frac{48\beta \lambda'^2_h \log({k_1}+1)}{\eta_{\min}^2} \sum_{k=k_0+1}^{{k_1}+1} \frac{1}{k} + + + \frac{48\beta\eta_{\max}^2 \log ({k_1}+1)}{\eta_{\min}^2} \sum_{k=k_0+1}^{{k_1}+1} \frac{\|y_{k-1}\|^2}{k} +2 \mu' \nonumber\\ + & \le \frac{96\beta \lambda'^2_h \log^2({k_1}+1)}{\eta_{\min}^2} + + + \frac{48\beta\eta_{\max}^2 \log ({k_1}+1)}{\eta_{\min}^2} \sum_{k=k_0}^{{k_1}} \frac{\|y_{k}\|^2}{k+1} +2 \mu' \nonumber\\ + & \le \frac{96\beta \log^2({k_1}+1)}{\eta_{\min}^2} \left( \lambda'^2_h + \eta_{\max}^2\sum_{k=k_0}^{{k_1}} \frac{\|y_{k}\|^2}{k+1} \right) + + +2 \mu', \label{eq:final bnd on G} \end{align} or, equivalently, \begin{align} -\min_{k_0 \le k \le K} \gamma_k \|G_k\|^2 & \le \frac{24\lambda'^2_h }{\eta_{\min}^2} M_{K} -+ \frac{24\eta_{\max}^2}{\eta_{\min}^2} M_{K}' -+ \frac{4\mu}{K-k_0+1}, +\min_{k \in K} \gamma_k \|G_k\|^2 & \le + \frac{192\beta \log^2({k_1}+1)}{\eta_{\min}^2|K|} \left( \lambda'^2_h + \eta_{\max}^2 |K|B_K \right) ++ \frac{4\mu'}{|K|}, \label{eq:bnd on gamma G} \end{align} where \begin{align} -M_K := \frac{\sum_{k=k_0+1}^{K+1} \sqrt{k}\beta_k}{K-k_0+1} , -\qquad -M_K' := \frac{\sum_{k=k_0+1}^{K+1} \sqrt{k}\beta_k \|y_{k-1}\|^2 }{K-k_0+1}. +\mu' := \frac{24\beta \log(k_1+1) \|G_{k_1+1}\|^2 }{\eta_{\min}^2 (k_1+1)} + \mu, +\label{eq:defn of muprime} \end{align} -%where the infinite series above converges, thanks to \eqref{eq:cnd on sigma gamma}. -%By our assumption in \eqref{eq:cnd on sigma gamma}, the series in the last line above is convergent and we find a final bound on the gradient mapping, namely -%\begin{align} -%\sum_{k=k_0}^\infty \gamma_k \|G_k\|^2 & -%\le \sum_{k=k_0+1}^\infty \frac{96\eta_{\max}^2 \sigma_k}{\eta_{\min}^2} \|y_{k-1}\|^2 + 2\mu' =: \mu''. -%\label{eq:final bnd on G} -%\end{align} +\begin{align} +B_{K} := \frac{1 }{|K|} \sum_{k=k_0}^{{k_1}} \frac{\|y_{k}\|^2}{k+1}, +\label{eq:defn of B} +\end{align} +and we will later estimate both $\mu',B_K$. In turn, the bound above on the gradient mapping controls the feasibility gap, namely, \begin{align} -& (K-k_0+1) \min_{k_0+1\le k \le K+1} \frac{\sqrt{k}\|A(u_k)-b\|^2}{\beta_k} \nonumber\\ -& \le \sum_{k=k_0+1}^{K+1} \frac{\sqrt{k}\|A(u_k)-b\|^2}{\beta_k} -\nonumber\\ -& \le \frac{3\lambda'^2_h}{\eta_{\min}^2} \sum_{k=k_0+1}^{K+1} \sqrt{k}\beta_k -+ \frac{3\eta_{\max}^2}{\eta_{\min}^2} \sum_{k=k_0+1}^{K+1} \sqrt{k}\beta_k \|y_{k-1}\|^2 + -\sum_{k=k_0+1}^{K+1} \frac{3\sqrt{k}\beta_k}{\eta_{\min}^2} \|G_k\|^2 -\qquad \text{(see \eqref{eq:to be used for feas})} -\nonumber\\ -& -\le \frac{3\lambda'^2_h(K-k_0+1)}{\eta_{\min}^2} M_k -+ \frac{3\eta_{\max}^2 (K-k_0+1)}{\eta_{\min}^2} M'_k + -\sum_{k=k_0+1}^{K+1} \frac{\gamma_k \|G_k\|^2 }{8} -\qquad \text{(see \eqref{eq:to be used later on})} -\nonumber\\ -& \le \frac{3\lambda'^2_h(K-k_0+1)}{\eta_{\min}^2} (M_k+M_{k+1}) -+ \frac{3\eta_{\max}^2 (K-k_0+1)}{\eta_{\min}^2} (M'_k+M'_{k+1}) + \frac{\mu}{2}, -\qquad \text{(see \eqref{eq:final bnd on G})} +& |K| \min_{k-1\in K} \log k \|A(u_k)-b\|^2 \nonumber\\ +& \le \sum_{k=k_0+1}^{{k_1}+1} \log k \|A(u_k)-b\|^2 \nonumber\\ +& \le \frac{12\beta^2 \lambda'^2_h}{\eta_{\min}^2} \sum_{k=k_0+1}^{{k_1}+1} \frac{\log k}{k} ++ \frac{12\beta^2 \eta_{\max}^2}{\eta_{\min}^2} \sum_{k=k_0+1}^{{k_1}+1} \frac{\|y_{k-1}\|^2 \log k}{k} ++ \sum_{k=k_0+1}^{{k_1}+1} \frac{12 \beta^2 \log k}{\eta_{\min}^2 k} \|G_k\|^2 +\qquad +\text{(see \eqref{eq:to be used for feas})} +\nonumber\\ +& \le \frac{24\beta^2 \lambda'^2_h \log^2 ({k_1}+1) }{\eta_{\min}^2 } ++ \frac{12\beta^2 \eta_{\max}^2 \log ({k_1}+1)}{\eta_{\min}^2} \cdot |K| B_{K} ++ \frac{\beta}{8}\sum_{k=k_0}^{{k_1}} \gamma_k \|G_k\|^2 ++ \frac{12\beta^2 \log (k_1+1) \|G_{k_1+1}\|^2 }{\eta_{\min}^2 (k_1+1)} +\qquad \text{(see (\ref{eq:to be used later on},\ref{eq:defn of B}))} \nonumber\\ +& \le + \frac{24\beta^2 \log^2(k_1+1)}{\eta_{\min}^2} \left( \lambda'^2_h+ \eta_{\max}^2 |K|B_K \right) ++ +\frac{24\beta^2 \log^2(k_1+1) }{\eta_{\min}^2} \left( \lambda'^2_h+ \eta_{\max}^2 |K|B_K \right) + \frac{\beta \mu'}{2} + \frac{12\beta^2 \log (k_1+1) \|G_{k_1+1}\|^2 }{\eta_{\min}^2 (k_1+1)} +\qquad \text{(see \eqref{eq:final bnd on G})} \nonumber\\ +& \le + \frac{24\beta^2 \log^2(k_1+1)}{\eta_{\min}^2} \left( \lambda'^2_h+ \eta_{\max}^2 |K|B_K \right) ++ +\frac{24\beta^2 \log^2(k_1+1) }{\eta_{\min}^2} \left( \lambda'^2_h+ \eta_{\max}^2 |K|B_K \right) + \beta \mu' +\qquad \text{(see \eqref{eq:defn of muprime})} + \nonumber\\ +& = + \frac{48\beta^2 \log^2(k_1+1)}{\eta_{\min}^2} \left( \lambda'^2_h+ \eta_{\max}^2 |K|B_K \right) + \beta\mu', \label{eq:final bnd on feas gap} \end{align} -or, equivalently, +which in turn implies that \begin{align} -\min_{k_0+1\le k \le K+1} \frac{\sqrt{k}\|A(u_k)-b\|^2}{\beta_k} & -\le \frac{3\lambda'^2_h}{\eta_{\min}^2} (M_k+M_{k+1}) -+ \frac{3\eta_{\max}^2 }{\eta_{\min}^2} (M'_k+M'_{k+1})+ \frac{\mu}{2(K-k_0+1)}. +\min_{k-1\in K} \|A(u_k)-b\|^2 & +\le \frac{48\beta^2 \log^2(k_1+1)}{\eta_{\min}^2|K|} \left( \lambda'^2_h+ \eta_{\max}^2 |K|B_K \right) + \frac{\beta\mu'}{|K|}, \label{eq:feas bnd semi final} \end{align} -%A useful consequence of \eqref{eq:final bnd on feas gap} is that -%\begin{align} -%o(1) & = \frac{\sqrt{k}\|A(u_k)\|^2}{\sigma_k} -%\qquad \text{(see \eqref{eq:final bnd on feas gap})} -%\nonumber\\ -%& = \frac{\sqrt{k}\|A(u_k)\|^2}{o(k^{-\frac{3}{2}})} -%\qquad \text{(2nd condition in \eqref{eq:recall 2nd major cnds})} \nonumber\\ -%& = o(k^2 \|A(u_k)\|^2), -%\label{eq:asymp feas gap} -%\end{align} -%namely, the feasibility gap decays as $\|A(u_k)\| = o(k^{-1})$. -%Here is the summary of conditions used: -%\begin{equation*} -%\beta_{k+1} \le \beta_k, -%\end{equation*} -%\begin{equation*} -%\sigma_k \le \frac{\eta_{k,\min}^2 \gamma_k}{288}, -%\qquad \sigma_k \le \frac{\beta_k}{2}, -%\end{equation*} -%\begin{equation*} -%\sigma_k = \frac{o(1) \eta_{k,\min}^2 }{k\eta_{k,\max}^2 \|y_{k-1}\|^2 }, -%\qquad \text{(Enforced in the algorithm)}. -%\end{equation*} -%\begin{equation} -%0< \eta_{k,\min}\|v\| \le \|P_{k+1}DA_k^\top v\|, -%\qquad \forall v. -%\label{eq:conditions} -%\end{equation} +if $k_0\ge 2$. Let us now revisit and simplify the condition imposed in (\ref{eq:good neighb}). -We first derive a uniform bound on the feasibility gap as +To that end, we first derive a weaker but uniform bound on the feasibility gap. For every $k-1\in K$, it holds that \begin{align} -\|A(u_k)-b\|^2 & \le \frac{\beta_k}{\sqrt{k}} \sum_{i=k_0}^k \frac{\sqrt{i}\|A(u_i)-b\|^2}{\beta_i} \nonumber\\ -& \le \frac{\beta_k}{\sqrt{k}} \left( \frac{3\lambda'^2_h k}{\eta_{\min}^2} (M_k +M_{k+1})+ \frac{3\eta_{\max}^2 k}{\eta_{\min}^2} (M'_k+M'_{k+1})+ \frac{\mu}{2} \right) -\qquad \text{(see \eqref{eq:final bnd on feas gap})} - \nonumber\\ -& = \sqrt{k}\beta_k \left( \frac{3\lambda'^2_h}{\eta_{\min}^2} (M_k+M_{k+1}) + \frac{3\eta_{\max}^2}{\eta_{\min}^2} (M'_k+M'_{k+1})+ \frac{\mu}{2k} \right) +\|A(u_k)-b\|^2 & \le \sum_{i=k_0+1}^{k_1+1} \log i \|A(u_i)-b\|^2 +\qquad \left( k_0\ge 2 \right) \nonumber\\ -& \le \frac{6\lambda_h'^2}{\eta_{\min}^2} \sqrt{k_0}\beta_{k_0}M_{k_0} + \frac{6\eta_{\max}^2 }{\eta_{\min}^2} \sqrt{k_0}\beta_{k_0}M'_{k_0} +\frac{\beta_{k_0} \mu}{4\sqrt{k_0}}, -\qquad \text{(see \eqref{eq:defn of M M'})} -% \nonumber\\ -%& \le \frac{6\lambda_h'^2M}{\eta_{\min}^2} + \frac{6\eta_{\max}^2 M'}{\eta_{\min}^2} + \frac{\beta_{k_0}\mu}{4\sqrt{k_0}}, -\qquad \text{(see \eqref{eq:sigma dec})} +& \le \frac{48\beta^2 \log^2(k_1+1)}{\eta_{\min}^2 } \left( \lambda'^2_h+ \eta_{\max}^2 |K|B_K \right) + \beta\mu'. +\qquad \text{(see \eqref{eq:feas bnd semi final})} \label{eq:rate of feas gap} \end{align} -%\begin{align} -%\sup_{k\ge k_0} \|A(u_k)\|^2 & \le \sup_{k\ge k_0} \frac{\sigma_k}{\sqrt{k}} \cdot \sup_{k\ge k_0} \frac{\sqrt{k}\|A(u_k)\|^2}{\sigma_k} \nonumber\\ -%& \le \sup_{k\ge k_0} \frac{\sigma_k}{\sqrt{k}} \cdot \sum_{k=k_0}^\infty \frac{\sqrt{k}\|A(u_k)\|^2}{\sigma_k} \nonumber\\ -%& \le \sup_{k\ge k_0} \frac{\sigma_k}{\sqrt{k}} \cdot \frac{\mu'}{4} -%\qquad \text{(see \eqref{eq:final bnd on feas gap})} \nonumber\\ -%& = \frac{\sigma_{k_0}}{\sqrt{k_0}} \cdot \frac{\mu'}{4}. -%\qquad \text{(see \eqref{eq:sigma dec})} -%%& \le \frac{4}{k_0^2} \cdot \frac{\mu'}{4} -%%\qquad \left( \text{1st condition in \eqref{eq:cnd on sigma gamma} and } k_0 \text{ sufficiently large} \right) \nonumber\\ -%%& = \frac{\mu'}{k_0^2}. -%\end{align} -where we assumed in the last line above that -\begin{align} -\label{eq:defn of M M'} -\{\sqrt{k}\beta_k M_k \}_{k\ge k_0}, -\qquad -\{\sqrt{k}\beta_k M_k' \}_{k \ge k_0}, -\end{align} -are both nonincreasing sequences. -Therefore, \eqref{eq:good neighb} would follow from the assumptions that +Therefore, we may replace \eqref{eq:good neighb} with the assumption that \begin{equation} \|A(u_{k_0})-b\| \le \rho, \qquad -\sqrt{k_0}\beta_{k_0}M_{k_0} \le \frac{ \eta_{\min}^2 \rho^2}{18 \lambda'^2_h}, -\qquad -\sqrt{k_0}\beta_{k_0}M'_{k_0} \le \frac{\eta_{\min}^2 \rho^2}{18\eta_{\max}^2 }, -\qquad -\beta_{k_0} \le \frac{4\sqrt{k_0}\rho^2}{3\mu}. +\frac{48\beta^2 \log^2(k_1+1)}{\eta_{\min}^2} \left( \lambda'^2_h+ \eta_{\max}^2 |K|B_K \right) + \beta\mu' \le \rho^2, \label{eq:suff cnd on rho} \end{equation} -%where the second assumption above is met for a sufficiently large $k_0$, thanks to the first assumption in \eqref{eq:cnd on sigma gamma}. -%For \eqref{eq:cnd for well cnd}, we write that -%\begin{align} -%4 \tau_C^2 \sup_{k\ge k_0}\gamma_k^2 \|G_k\|^2 -%& \le 4 \tau_C^2 \sup_{k\ge k_0}\gamma_k \|G_k\|^2 \cdot \sup_{k\ge k_0} \gamma_k \nonumber\\ -%& \le 4 \tau_C^2 \sum_{k=k_0}^\infty \gamma_k \|G_k\|^2 \cdot \sup_{k\ge k_0} \gamma_k\nonumber\\ -%& \le 4\tau_C^2 \mu' \sup_{k\ge k_0} \gamma_k. -%\qquad \text{(see \eqref{eq:final bnd on G})} -%\end{align} -%Therefore, \eqref{eq:cnd for well cnd} holds if -%\begin{equation} -%\frac{\eta_{\min}}{2\lambda_A} \ge 4 \mu' \sup_{k\ge k_0} \gamma_k = 4\mu' \gamma_{k_0}, -%\end{equation} -%where the identity holds thanks to \eqref{eq:sigma dec}. -It remains to collect all the conditions so far on the sequences $\{\beta_k,\sigma_k\}_k$. It is easy to verify that (\ref{eq:beta n sigma},\ref{eq:to be used later on}) follow from the assumptions +which ensures that \begin{align} -\beta_k \le \sigma_k \wedge \frac{\eta_{\min}^2 \gamma_k}{24\sqrt{k}}, +\|A(u_k)-b\| \le \rho, +\qquad \forall k\in [k_0:k_1+1]. +\label{eq:uniform feas larger interval} \end{align} +In order to interpret (\ref{eq:final bnd on G},\ref{eq:feas bnd semi final},\ref{eq:suff cnd on rho}), we next estimate $B_{K}$ in \eqref{eq:defn of B}. To that end, let us first control the growth of the dual sequence $\{y_k\}_k$. Recalling \eqref{eq:y update recall} and for every $k\in [k_0:k_1+1]$, we write that \begin{align} -\beta_{k_0}\le \frac{\eta_{\min} \rho }{\sqrt{18(k_0+1)}\lambda'_h} -\wedge -\frac{\eta_{\min} \rho }{\sqrt{18(k_0+1)}\eta_{\max} \|y_{k_0}\|} -\wedge -\frac{4\sqrt{k_0}\rho^2}{3\mu}, +\|y_k\| & \le \|y_{k_0}\|+ \sum_{i=k_0+1}^k \frac{\|A(u_i)-b\|}{\sigma_k} +\qquad \text{(see \eqref{eq:y update recall})} +\nonumber\\ +& \le \|y_{k_0}\|+ \sum_{i=k_0+1}^k \frac{\rho}{\beta k} +\qquad \text{(see (\ref{eq:beta n sigma},\ref{eq:uniform feas larger interval}))} \nonumber\\ +& \le \|y_{k_0}\|+ \frac{2\rho \log k}{\beta}. +\label{eq:dual growth} \end{align} -where $\wedge$ stands for minimum. Let us also revisit \eqref{eq:defn of M M'}. Using \eqref{eq:sigma dec} and after some algebraic manipulations, we may verify that $\{\sqrt{k}\beta_kM_k\}_k$ and $\{\sqrt{k}\beta_kM'_k\}_k$ are nonincreasing sequences if +With the growth of the dual sequence uncovered, we evaluate $B_{K}$ as \begin{align} -\left\{\frac{\sum_{i=k_0+1}^{k+1} \sqrt{i}\beta_i}{\sqrt{k-k_0+1}}\right\}_{k\ge k_0}, +B_{K} & = \frac{1}{|K|}\sum_{k=k_0}^{{k_1}} \frac{ \|y_{k}\|^2 }{k+1} +\qquad \text{(see \eqref{eq:defn of B})} +\nonumber\\ +& \le \frac{1}{|K|} \sum_{k=k_0}^{{k_1}} \frac{1}{k+1}\left( \|y_{k_0}\|+ \frac{2\rho \log k}{\beta} \right)^2 +\qquad \text{(see (\ref{eq:dual growth}))} +\nonumber\\ +& \le \frac{1}{|K|} \sum_{k=k_0}^{{k_1}} \frac{2\|y_{k_0}\|^2}{k+1}+ \frac{8\rho^2 \log^2 k}{\beta^2 (k+1)} +\qquad ((a+b)^2 \le 2a^2+2b^2) +\nonumber\\ +& \le \frac{4\|y_{k_0}\|^2 \log ({k_1}+1)}{|K|} + \frac{16 \rho^2 \log^3 ({k_1}+1)}{|K|} \nonumber\\ +& \le \frac{16 \log^3(k_1+1)}{ |K|} \left( \|y_{k_0}\|^2+\rho^2 \right). +\label{eq:Bk evaluated} +\end{align} +In order to interpret (\ref{eq:final bnd on G},\ref{eq:feas bnd semi final},\ref{eq:suff cnd on rho}), it still remains to estimate $\mu'$ in \eqref{eq:defn of muprime}. To that end, we first derive a lower bound on the step sizes $\{\gamma_k\}_k$. To invoke \eqref{eq:moreover}, we in turn need to gauge how smooth the augmented Lagrangian $\mathcal{L}_{\beta_k}(\cdot,y_k)$ is. For every $k\in [k_0:k_1+1]$, note that +\begin{align} +\lambda_{\beta_k} & \le \lambda_h+ \sqrt{m}\lambda_A \left( \|y_k\| + \frac{\|A(u_k)-b\|}{\beta_k} \right)+ \frac{\|DA(u_k)\|_F^2}{\beta_k} +\qquad \text{(see \eqref{eq:smoothness of Lagrangian})} \nonumber\\ +& \le \lambda_h+ \sqrt{m}\lambda_A \left( \|y_{k_0}\|+ \frac{2\rho \log (k_1+1)}{\beta_k} + \frac{\rho}{\beta_k} \right) + \frac{ m \eta_{\max}^2}{\beta_k} +\qquad \text{(see (\ref{eq:defn lambdap n etaMax},\ref{eq:uniform feas larger interval},\ref{eq:dual growth}))} \nonumber\\ +& = \lambda_h+\sqrt{m}\lambda_A \|y_{k_0}\| + \frac{ 1}{\beta_k}(2\sqrt{m}\lambda_A\rho \log(k_1+1) + \sqrt{m}\lambda_A \rho + m\eta_{\max}^2 ) \nonumber\\ +& \le \frac{ 2}{\beta_k}(2\sqrt{m}\lambda_A\rho \log(k_1+1) + \sqrt{m}\lambda_A \rho + m\eta_{\max}^2 ), +\label{eq:smoothness at k} +\end{align} +where the last line holds if $k_0$ is sufficiently large. We are now in position to invoke \eqref{eq:moreover} by writing that +\begin{align} +\gamma_k & \ge \frac{\theta}{\lambda_{\beta_k}} \qquad \text{(see \eqref{eq:moreover})}\nonumber\\ +& \ge \frac{\theta \beta_k}{ 4\sqrt{m}\lambda_A\rho \log(k_1+1) + 2\sqrt{m}\lambda_A \rho + 2m\eta_{\max}^2 } +\qquad \text{(see \eqref{eq:smoothness at k})} \nonumber\\ +& = \frac{\theta \beta}{ (4\sqrt{m}\lambda_A\rho \log(k_1+1) + 2\sqrt{m}\lambda_A \rho + 2m\eta_{\max}^2 ) \sqrt{k} }, +\qquad \text{(see \eqref{eq:beta n sigma})} +\label{eq:low bnd on gammas} +\end{align} +for every $k\in [k_0:k_1+1]$. +In particular, the lower bound on $\gamma_{k_1+1}$ above allows us to estimate $\mu'$ by writing that +\begin{align} +\mu' & = \frac{24\beta \log(k_1+1) \|G_{k_1+1}\|^2 }{\eta_{\min}^2 (k_1+1)} + \mu +\qquad \text{(see \eqref{eq:defn of muprime})} \nonumber\\ +& = \frac{24\beta \log(k_1+1) \|u_{k_1+2}-u_{k_1+1}\|^2 }{\eta_{\min}^2 (k_1+1) \gamma_{k_1+1}^2} + \mu +\qquad \text{(see \eqref{eq:grad map recall})} \nonumber\\ +& \le + \frac{24\beta \log(k_1+1) \text{diam}(C)^2 }{\eta_{\min}^2 (k_1+1) \gamma_{k_1+1}^2} + \mu + \qquad \left( \{u_k\} \subset C \right) + \nonumber\\ + & \le + \frac{24\beta \log(k_1+1) \text{diam}(C)^2 (4\sqrt{m}\lambda_A\rho \log(k_1+1) + 2\sqrt{m}\lambda_A \rho + 2m\eta_{\max}^2 )^2 }{\eta_{\min}^2 \theta^2 \beta^2} ++2\mu +\qquad \text{(see \eqref{eq:low bnd on gammas})} +\nonumber\\ +& =: \mu''. +\label{eq:mup to mupp} +\end{align} +Having estimated $B_K$ and $\mu'$, we can rewrite the bound on the feasibility gap as +\begin{align} +\min_{k-1\in[K]}\|A(u_k)-b\|^2 +& \le \frac{48\beta^2 \log^2(k_1+1)}{\eta_{\min}^2|K|} \left( \lambda'^2_h+ \eta_{\max}^2 |K|B_K \right) + \frac{\beta\mu'}{|K|} +\qquad \text{(see \eqref{eq:feas bnd semi final})} \nonumber\\ +& \le \frac{48\beta^2 \log^2(k_1+1)}{\eta_{\min}^2|K|} \left( \lambda'^2_h+ 16 \eta_{\max}^2 (\|y_{k_0}\|^2 +\rho^2) \log^3(k_1+1) \right) + \frac{\beta\mu''}{|K|} +\qquad \text{(see (\ref{eq:Bk evaluated},\ref{eq:mup to mupp}))} +\end{align} +Moreover, we can simplify the assumption in \eqref{eq:suff cnd on rho}. To be specific, thanks to (\ref{eq:Bk evaluated},\ref{eq:mup to mupp}), we can replace \eqref{eq:suff cnd on rho} with the assumption +\begin{align} +\|A(u_{k_0})-b\|\le \rho, \qquad -\left\{\frac{\sum_{i=k_0+1}^{k+1} \sqrt{i}\beta_i \|y_{i-1}\|^2}{\sqrt{k-k_0+1}}\right\}_{k\ge k_0}, -\label{eq:sequence grows slowly} +\frac{48\beta^2 \log^2(k_1+1)}{\eta_{\min}^2} \left( \lambda'^2_h+ 16 \eta_{\max}^2 (\|y_{k_0}\|^2+\rho^2) \log^3(k_1+1) \right) + \beta\mu'' \le \rho^2. +\label{eq:suff cnd 2nd time} \end{align} -are both nonincreasing sequences. In particular, observe that the assumptions in \eqref{eq:sequence grows slowly} imply that $\{M_k,M'_k\}_{k\ge k_0}$ are both nonincseasing sequences as well, which in turn simplifies the bound on the feasibility in (\ref{eq:feas bnd semi final}) as +The lower bound on the step sizes in \eqref{eq:low bnd on gammas} has two other consequences. First, we find that \eqref{eq:to be used later on} automatically holds if $k_0$ is sufficiently large. Second, it allows us to improve \eqref{eq:bnd on gamma G} by writing that \begin{align} -\min_{k_0 \le i \le k} \frac{\sqrt{k}\|A(u_k)-b\|^2}{\beta_k} \le \frac{6\lambda'^2_h}{\eta_{\min}^2} M_k + \frac{6\eta_{\max^2}}{\eta_{\min}^2} M'_k+ \frac{\mu}{2(k-k_0+1)}. -\end{align} -%To simplify the above conditions, first note that -%\begin{align} -% \frac{\sigma_k}{8\|A(u_{k})\|} & -%\ge \frac{\sigma_k}{8\sup_{k'\ge k }\|A(u_{k'})\|} \nonumber\\ -%& \ge \frac{ k^{\frac{1}{4}}\sqrt{\sigma_k}}{4\sqrt{\mu'}} , -%\qquad \text{(see \eqref{eq:rate of feas gap})} -%\end{align} -%which allows us to tighten the conditions in (\ref{eq:recall 2nd major cnds},\ref{eq:recall smoothness cnd on gamma}) as -%\begin{align} -%\sigma_k \le \beta_k, -%\quad -%\sigma_k = o(\sqrt{k}), -%\quad -%\frac{\sqrt{k}\sigma_k}{\gamma_k} = o(1), -%\quad -%k^{\frac{3}{2}} \sigma_k \|y_{k-1}\|^2 =o(1), -%\end{align} -%\begin{align} -%\gamma_k \le \frac{1}{4\lambda_h} \wedge \frac{\eta_{\min}}{8\lambda_A\mu'}, -%\qquad -%\gamma_k \le \frac{1}{4m\lambda_A \|y_{k-1}\|}, -%\qquad -%\gamma_k \le \frac{k^{\frac{1}{4}}\sqrt{\sigma_k}}{4\sqrt{\mu'}}, -%\qquad -%\gamma_k \le \frac{\beta_k}{4m\eta_{\max}^2}, -%\qquad \forall k \ge k_0. -%\end{align} -%It suffices to take $\beta_k=\beta$ to be constant for all $k$. Also, the above relation between $\{\sigma_k,\gamma_k\}_k$ can be tightened to -%\begin{align} -%\sigma_k = o(k^{-\frac{3}{2}}), -%\qquad -%\frac{\sqrt{k} \sigma_k}{o(1)} = \gamma_k = o(k) \sigma_k. -%\end{align} -We summarize our findings below. -%In particular, if the dual sequence $\{y_k\}$ converges, we can derive a convergence rate. We can take $\sigma_k = k^{-\frac{3}{2}-\epsilon}$ and $\gamma_k = k^{-\frac{1}{2}-2\epsilon}$ and $\beta_k = \beta$. Then -%$$ -%\sum_{k=k_0}^\infty \frac{\sqrt{k}\|A(u_k)\|^2}{\sigma_k} < \infty -%\Longrightarrow \|A(u_k)\| = o(k^{-\frac{3}{2}-\epsilon}), -%$$ -%$$ -%\sum_{k=k_0}^\infty \gamma_k \|G_k\|^2 <\infty -%\Longrightarrow -%\|G_k\|^2 = o(k^{-\frac{1}{2}+\epsilon}). -%$$ - - - - -%Our argument is as follows. We select sequences $\{\beta_k,\sigma_k,\gamma_k\}_k$ that satisfy \eqref{eq:recall 2nd major cnds}. -%Then we multiply both sides of \eqref{eq:recall smoothness cnd on gamma} by $\sqrt{k}\sigma_k$ and apply the third condition in \eqref{eq:recall 2nd major cnds} to find that -%\begin{align} -%o(1) = \frac{\sqrt{k}\sigma_k}{\gamma_k} & \ge \sqrt{k}\sigma_k \lambda_h -%+ \sqrt{m}\lambda_A \cdot \sqrt{k}\sigma_k \|y_{k-1}\| -%+ 2\sqrt{m}\lambda_A \cdot \sqrt{k} \|A(u_k)\| + \frac{\sqrt{k}\sigma_k\|DA(u_k)\|^2}{\beta_k} -%\qquad \text{(3rd condition in \eqref{eq:recall 2nd major cnds})} \nonumber\\ -%& = o(1) + 2\sqrt{m}\lambda_A \cdot \sqrt{k} \|A(u_k)\| + \frac{\sqrt{k}\sigma_k\|DA(u_k)\|^2}{\beta_k}. -%\qquad \text{(2nd and 4th conditions in \eqref{eq:recall 2nd major cnds})} \nonumber\\ -%& = o(1) + \frac{\sqrt{k}\sigma_k\|DA(u_k)\|^2}{\beta_k}. -%\qquad \text{(see \eqref{eq:asymp feas gap})} -%\end{align} -%Therefore, \eqref{eq:recall smoothness cnd on gamma} is asymptotically equivalent to -%\begin{align} -%\frac{\sqrt{k}\sigma_k}{\beta_k} = o(1), -%\end{align} -%which does not contradict \eqref{eq:recall 2nd major cnds}. To summarize, the conditions on the sequences $\{\beta_k,\sigma_k,\gamma_k\}$ are -%\begin{align} -%\sigma_{k+1} \le \sigma_k \le \beta_k, -%\qquad -% k^{\frac{3}{2}}\sigma_k = o(1), -%\qquad -%\frac{\sqrt{k}\sigma_k}{\gamma_k} = o(1), -%\end{align} -%\begin{align} -%\sqrt{k}\sigma_k \|y_{k-1}\| = o(1), -%\qquad -%k^{\frac{3}{2}} \sigma_k \|y_{k-1}\|^2 =o(1), -%\qquad -%\gamma_k = o(1), -%\end{align} -%\begin{align} -%\frac{1}{\gamma_k} -%& \ge \lambda_h + \sqrt{m}\lambda_A \left(\|y_{k-1}\|+ \frac{2\|A(u_k)\|}{\sigma_k} \right) + \frac{\|DA(u_k)\|_F^2}{\beta_k}. -%\end{align} - +& \min_{ k\in K } \| G_k\|^2 \nonumber\\ +& \le \frac{\min_{k \in K} \gamma_k \|G_k\|^2}{\min_{k\in K} \gamma_k } \nonumber\\ +& \le \frac{1}{\min_{k\in K} \gamma_k} +\left( \frac{192\beta \log^2({k_1}+1)}{\eta_{\min}^2|K|} \left( \lambda'^2_h + \eta_{\max}^2 |K|B_K \right) ++ \frac{4\mu'}{|K|} \right) +\qquad \text{(see \eqref{eq:bnd on gamma G})} + \nonumber\\ +& \le \frac{1}{\min_{k\in K} \gamma_k} +\left( \frac{192\beta \log^2({k_1}+1)}{\eta_{\min}^2|K|} \left( \lambda'^2_h + 16 \eta_{\max}^2 (\|y_{k_0}\|^2+\rho^2) \log^3 (k_1+1) \right) ++ \frac{4\mu''}{|K|} \right) +\qquad \text{(see (\ref{eq:Bk evaluated},\ref{eq:mup to mupp}))} + \nonumber\\ + & \le \frac{ (4\sqrt{m}\lambda_A\rho \log(k_1+1) + 2\sqrt{m}\lambda_A \rho + 2m\eta_{\max}^2 ) \sqrt{k}}{\theta \beta} +\left( \frac{192\beta \log^2({k_1}+1)}{\eta_{\min}^2|K|} \left( \lambda'^2_h + 16 \eta_{\max}^2 (\|y_{k_0}\|^2+\rho^2) \log^3 (k_1+1) \right) ++ \frac{4\mu''}{|K|} \right) +\quad \text{(see \eqref{eq:low bnd on gammas})} \nonumber\\ +& \le \frac{ 4\sqrt{m}\lambda_A\rho \log(k_1+1) + 2\sqrt{m}\lambda_A \rho + 2m\eta_{\max}^2 }{c\theta \beta \sqrt{k_1}} +\left( \frac{192\beta \log^2({k_1}+1)}{\eta_{\min}^2 } \left( \lambda'^2_h + 16 \eta_{\max}^2 (\|y_{k_0}\|^2+\rho^2) \log^3 (k_1+1) \right) ++ 4\mu'' \right), +\end{align} +where the last line holds if there exists $c>0$ for which $k_1-k_0+1 \ge ck_1 $. \begin{theorem} \label{thm:main} Let $\gamma_k$ be the output of the line search subroutine in our algorithm in iteration $k$. -For an integer $k_0$, suppose that the sequences $\{\beta_k,\sigma_k\}_k$ satisfy -\begin{align} -\beta_k \le \sigma_k \wedge \frac{\eta_{\min}^2 \gamma_k}{24\sqrt{k}}, -\end{align} +For integers $k_0\le k_1$, consider the interval $K=[k_0:k_1]$ and suppose that \begin{align} -\beta_{k_0} \le \frac{\eta_{\min} \rho }{\sqrt{18(k_0+1)}\lambda'_h} -\wedge -\frac{\eta_{\min} \rho }{\sqrt{18(k_0+1)}\eta_{\max} \|y_{k_0}\|} -\wedge -\frac{4\sqrt{k_0}\rho^2}{3\mu}, -\end{align} -where $\wedge$ stands for minimum and $\lambda'_h = \max_{u\in C}\|\nabla h(u)\|$. Moreover, $\{\beta_k\}$ is selected so that both sequences -\begin{align} -\left\{\frac{\sum_{i=k_0+1}^{k+1} \sqrt{i}\beta_i}{\sqrt{k-k_0+1}}\right\}_{k\ge k_0}, +\beta_k = \frac{\beta}{\sqrt{k}}, \qquad -\left\{\frac{\sum_{i=k_0+1}^{k+1} \sqrt{i}\beta_i \|y_{i-1}\|^2}{\sqrt{k-k_0+1}}\right\}_{k\ge k_0}, +\sigma_k = \beta k, +\qquad \forall k\in K. \end{align} -are nonincreasing. We impose the following geometric requirements on the constraints. Let $P_{T_C(u)}$ and $P_{N_C(u)}$ denote the orthogonal projection onto the tangent and normal cones at $u\in C$, respectively. -Consider a subspace $S_{k_0}\subseteq \mathbb{R}^d$ such that +We impose the following geometric requirements on the constraints. Let $P_{T_C(u)}$ and $P_{N_C(u)}$ denote the orthogonal projection onto the tangent and normal cones at $u\in C$, respectively. +Consider a subspace $S_{K}\subseteq \mathbb{R}^d$ such that \begin{align} -S_{k_0} \supseteq \bigcup_{k\ge k_0} T_C(u_k), +S_{K} \supseteq \bigcup_{k\in K} T_C(u_k), \end{align} -and, with some abuse of notation, let $S_{k_0}$ also denote an orthonormal basis for this subspace. For $\eta_{\min},\rho>0$, we assume that the nonconvex Slater's condition holds, namely, +and, with some abuse of notation, let $S_{K}$ also denote an orthonormal basis for this subspace. For $\eta_{\min}$, we assume that the nonconvex Slater's condition holds, namely, that there exists $\rho>0$ such that \begin{align} 0 < \eta_{\min} := \begin{cases} \min_u \, \left\| S_{k_0}^\top P_{T_C(u)} (DA(u)^\top v) \right\| \\ \|v\|=1\\ \|A(u)-b\|\le \rho\\ u\in C. \end{cases} \label{eq:new slater} \end{align} -Suppose that $\|A(u_{k_0})-b\| \le \rho$ and that $\operatorname{diam}(C)\le 2\eta_{\min}/\lambda_A$. +Suppose that +\begin{align} +\|A(u_{k_0})-b\| \le \rho, +\qquad +\operatorname{diag}(C) \le \frac{2\eta_{\min}}{\lambda_A}, +\qquad +\rho \ge + \rho_{\operatorname{low}} (C,A,\beta) \log^{\frac{5}{2}} k_1 , +\end{align} +where $\rho_{\operatorname{low}} (C,A,\beta)$ depends only on $C,A,\beta$ and is specified in the proof, see (\ref{eq:suff cnd 2nd time}). Then it holds that \begin{align} -\min_{k_0 \le i \le k} \gamma_k \|G_{\beta_i,\gamma_i}(u_i,y_i)\|^2 & \le \frac{24\lambda'^2_h }{\eta_{\min}^2} M_{k} -+ \frac{24\eta_{\max}^2}{\eta_{\min}^2} M_{k}' -+ \frac{4\mu}{k-k_0+1}, +\min_{k\in K} \|G_{\beta_i,\gamma_i}(u_i,y_i)\|^2 & = \frac{O(\log^5 k_1)}{\sqrt{k_1}}, +\qquad \forall k\in K, \end{align} \begin{align} -\min_{k_0 \le i \le k} \frac{\sqrt{k}\|A(u_k)-b\|^2}{\beta_k} \le \frac{6\lambda'^2_h}{\eta_{\min}^2} M_k + \frac{6\eta_{\max^2}}{\eta_{\min}^2} M'_k+ \frac{\mu}{2(k-k_0+1)}. +\min_{k\in K} \| A(u_k)-b\| = \frac{O(\log^5 k_1)}{\sqrt{k_1}}, +\qquad \forall k\in K, \end{align} -for every $k\ge k_0$, -provided that +provided that $k_0 = \Omega(k_1)$ is sufficiently large and \begin{equation} -\inf_k h(u_k) + \langle A(u_k) ,y_{k_0}\rangle > -\infty. +\inf_k h(u_k) + \langle A(u_k)-b ,y_{k_0}\rangle > -\infty. \end{equation} -Above, -\begin{align} -M_k = \frac{\sum_{i=k_0+1}^{k+1} \sqrt{i}\beta_i}{k-i+1}, -\qquad -M_k' = \frac{\sum_{i=k_0+1}^{k+1} \sqrt{i}\beta_i \|y_{i-1}\|^2}{k-i+1}. -\end{align} -A (weaker) bound also holds for the feasibility gap, namely, -\begin{align} -\|A(u_k)-b\|^2 \le -\frac{6\lambda_h'^2}{\eta_{\min}^2} \sqrt{k}\beta_k M_k + \frac{6\eta_{\max}^2}{\eta_{\min}^2} \sqrt{k}\beta_k M'_k+ \frac{\mu \beta_k}{2\sqrt{k}}, -\qquad \forall k\ge k_0. -\end{align} -%In particular, if the dual sequence $\{y_k\}_k$ converges, then we can take -%\begin{align} -%\beta_k = \beta, -%\qquad -%\sigma_k = \frac{\sigma_0}{k}. -%\end{align} -%Then, for sufficiently large $k_0$, -%it holds that -%\begin{align} -% \min_{k_0 \le i \le k} \gamma_i \|G_{\beta_i,\gamma_i}(u_i,y_i)\|^2 = \frac{O(1)}{\sqrt{k}}, -% \qquad -% \min_{k_0 \le i \le k} \|A(u_i)-b\| = \frac{O(1)}{k}. -%\end{align} -% -% -% - \end{theorem} -The special of Theorem \ref{thm:main}, presented below, is of particular interest. -\begin{corollary} -In Theorem \ref{thm:main}, let us take $\gamma_k$ to be the output of the line search routine in Lemma \ref{lem:eval Lipsc cte}. For an integer $k_0$, let $\sigma_k = \sigma$, and $\beta_k = \beta_{k_0} k^{-\frac{3}{2}}$ for every $k\ge k_0$ and a sufficiently small $\beta_{k_0}$. For positive $\rho,\eta_{\min}$, assume also that the Slater condition in (\ref{eq:new slater}) holds, that $\|A(u_{k_0})-b\|\le \rho$, and that $\operatorname{diam}(C)\le 2\eta_{\min}/\lambda_A$. Then it holds that -\begin{align} -\min_{k_0 \le i \le k} \gamma_k \|G_{\beta_i,\gamma_i} (u_i,y_i)\|^2 = \frac{O(\log k)}{k}, -\end{align} -\begin{align} -\min_{k_0 \le i \le k} k \|A(u_k)-b\| = \sqrt{\frac{O(\log k)}{k}}, -\end{align} -\begin{align} -\|A(u_k)-b\| = \frac{O(\sqrt{\log k})}{k}, -\qquad -\forall k \ge k_0, -\end{align} -provided that -\begin{align} -\inf_k h(u_k) + \langle A(u_k),y_{k_0} \rangle >-\infty. -\end{align} -\end{corollary} + %\paragraph{\textbf{Summary of Bang's argument.}} Bang's argument is summarized below for comparison: %\begin{align} %& \frac{\gamma_k G_k^2}{2} \nonumber\\ %& \le h_k - h_{k+1} + (A_k - A_{k+1}) \cdot y_k + \frac{\|A_k\|^2}{2\beta_k} - \frac{\|A_{k+1}\|^2}{2\beta_k} \nonumber\\ %& = h_k - h_{k+1} + A_k \cdot y_k - A_{k+1} \cdot y_{k+1} + A_{k+1} \cdot (y_{k+1} - y_{k}) + \frac{\|A_k\|^2}{2\beta_k} - \frac{\|A_{k+1}\|^2}{2\beta_k} \nonumber\\ %& = h_k - h_{k+1} + A_k \cdot y_k - A_{k+1} \cdot y_{k+1} + \left(\frac{1}{\sigma_{k+1}} - \frac{1}{2\beta_k} \right) \|A_{k+1}\|^2 + \frac{\|A_k\|^2}{2\beta_k} %\quad \text{(definition of ys)} % \nonumber\\ %& \le (h_k +A_k \cdot y_k + \frac{G_{k-1}}{4} + \frac{\|A_k\|^2}{2\beta_{k-1}}) - (h_{k+1}+ A_{k+1} \cdot y_{k+1} + \frac{G_k}{4} + \frac{\|A_{k+1}\|^2}{2\beta_k}) + e_k. %\qquad \text{(Bang's assumption)} %\end{align} %\textbf{We form a telescope with the last line above, which throws away any information in $\{y_k\}$. %} \paragraph{\textbf{Example: Max-cut}} Consider the factorized max-cut program, namely, \begin{align} \begin{cases} \min \langle UU^\top , H\rangle\\ \text{diag}(UU^\top) = 1, \end{cases} \end{align} where $U\in \mathbb{R}^{d'\times r}$. For every $i$, let $u_i\in\mathbb{R}^r$ denote the $i$th row of $U$. Let us form $u\in\mathbb{R}^d$ with $d=d'r$ by vectorizing $U$, namely, \begin{equation} u = [u_1^\top \cdots u_{d'}^\top]^\top. \end{equation} We can therefore cast the above program as Program \eqref{prob:01} with \begin{align} h(u) = \sum_{i,j} H_{i,j} \langle u_i, u_j \rangle, \end{align} \begin{align} A: u \rightarrow [\|u_1\|^2 \cdots \|u_{d'}\|^2]^\top. \end{align} It is easy to verify that \begin{align} DA(u) = \left[ \begin{array}{ccc} u_1^\top & \cdots & 0\\ \vdots\\ 0 & \cdots & u_{d'}^\top \end{array} \right]\in \mathbb{R}^{d'\times d}. \end{align} In particular, if we take $S_{k_0}=\mathbb{R}^d$ and $\rho <1$, we have $P_{T_C(u)}=I_d$ and thus \begin{align} \eta_{\min} & = \begin{cases} \min_u \eta_{\min}\left( DA(u) \right)\\ \|A(u) - 1\| \le \rho \end{cases} \nonumber\\ & = \begin{cases} \min_u \min_i \|u_i\|^2 \\ | \|u_i\|^2 - 1 | \le \rho \qquad \forall i \end{cases} \nonumber\\ & \ge 1- \rho >0. \end{align} Above, $\eta_{\min}(DA(u))$ returns the smallest singular value of $DA(u)$. Consequently, the nonconvex Slater's condition holds for the max-cut problem. \paragraph{\textbf{Example: Clustering}} Consider the factorized clustering problem, namely, \begin{align} \begin{cases} \min \langle UU^\top , H \rangle \\ UU^\top 1 = 1 \\ \|U\|_F \le \sqrt{k}\\ U \ge 0, \end{cases} \end{align} where $U\in \mathbb{R}^{d'\times r}$ and $k$ is the number of clusters. We form $u\in \mathbb{R}^d$ as before. Note that the above program can be cast as Program \eqref{prob:01} with the same $h$ as before and \begin{align} A: u\rightarrow \left[ \begin{array}{ccc} u_1^\top \sum_j u_j & \cdots u_{d'}^\top \sum_j u_j \end{array} \right]^\top \in \mathbb{R}^{d'}, \end{align} and also $C=\sqrt{k}B_{2+}$, where $B_{2+}\subset \mathbb{R}^{d'}$ is the intersection of the unit $\ell_2$-ball with the positive orthant. Note that \begin{align} DA(u) = \left[ \begin{array}{ccc} w_{1,1} u_1^\top & \cdots & w_{1,d'} u_{1}^\top\\ \vdots\\ w_{d',1} u_{d'}^\top & \cdots & w_{d',d'} u_{d'}^\top \end{array} \right], \label{eq:Jacobian clustering} \end{align} where $w_{i.i}=2$ and $w_{i,j}=1$ for $i\ne j$. Let us start with the case where $u\in \partial C$ belongs to the boundary of $C$, namely, $\|u\|=\sqrt{k}$. We also assume that $u>0$. Under these assumptions, note that \begin{equation} T_C(u)=\{z \in \mathbb{R}^{d'}: \langle u, z\rangle = 0 \}, \end{equation} and, consequently, \begin{equation} P_{T_C(u)} = I_d - \frac{uu^\top}{\|u\|^2} = I_d - \frac{uu^\top}{k}. \end{equation} For simplicity, let us assume that $\rho=0$, namely, $u$ is a feasible point of Program \eqref{prob:01}. Then we find that \begin{align} \eta_{\min} ( P_{T_C(u)} DA(u)^\top) & = \eta_{\min} ( (I-\frac{uu^\top}{k} ) DA(u)^\top) \nonumber\\ & \ge \eta_{\min} ( DA(u) ) - \frac{1}{k} \| uu^\top DA(u)^\top\| \qquad \text{(Weyl's inequality)} \nonumber\\ & = \eta_{\min} ( DA(u) ) - \frac{1}{\sqrt{k}}\| DA(u) u\| \qquad \left( \|u\|=\frac{1}{\sqrt{k}} \right). \label{eq:lower bnd} \end{align} We evaluate each term in the last line above separately. By its definition in \eqref{eq:Jacobian clustering}, first note that \begin{align} \eta_{\min}(DA(u)) & \ge \eta_{\min}\left( \left[ \begin{array}{ccc} u_1 & \cdots & u_{d'}\\ \vdots\\ u_1 & \cdots & u_{d'} \end{array} \right]\right) - \max_i \|u_i\| \qquad \text{(Weyl's inequality)} \nonumber\\ & = \sqrt{d}' \eta_{\min}(U) -\max_i \|u_i\| \nonumber\\ & \ge \sqrt{d'} \eta_{\min }(U) - 1, \label{eq:lower bnd on DA} \end{align} where in the last line follows from the assumptions that $u_i^\top \sum_j u_j =1$ for all $i$ and that $u>0$ to see that $\max_i \|u_i\|\le 1$. Note also that \begin{align} \|DA(u) u \| & \le \left\| \left[ \begin{array}{ccc} u_1^\top & \cdots & u_1^\top\\ \vdots\\ u_{d'}^\top & \cdots & u_{d'}^\top \end{array} \right] u \right\| + \sqrt{ \sum_{i=1}^{d'} \|u_i\|^4} \nonumber\\ & = \|1_{d'}\|+\max_i \|u_i\| \cdot \|u\| \nonumber\\ & \le \sqrt{d'}+ \sqrt{k}, \end{align} where the last line follows because $\max_i\|u_i\|\le 1$ and $\|u\|=\sqrt{k}$ by assumption. Consequently, we reach \begin{align} \eta_{\min} ( P_{T_C(u)} DA(u)^\top) \ge & \sqrt{d'} \left(\eta_{\min}(U) - \frac{1}{\sqrt{k}} \right) - 2 . \qquad \text{(see \eqref{eq:lower bnd})} \end{align} By continuity, we extend the results to the case where $u $ might have zero entries. Lastly, in the case where $u\in \text{int}(C)$ (namely, $\|u\|< \sqrt{k}$ and $u>0$), we have that $T_C(u)=\mathbb{R}^d$ and it directly follows from \eqref{eq:lower bnd on DA} that \begin{align} \eta_{\min} (P_{T_C(u)} DA(u)) & = \eta_{\min} (DA(u))\nonumber\\ & \ge \sqrt{d'} \eta_{\min}(U) -1. \qquad \text{(see \eqref{eq:lower bnd on DA})} \end{align} Having studied all cases for $u$ for $\rho=0$, we conclude that \begin{align} \eta_{\min} \ge \sqrt{d'} \left(\eta_{\min}(U) - \frac{1}{\sqrt{k}} \right) - 2. \qquad \text{(see \eqref{eq:new slater})} \end{align} Roughly speaking, as long as $\eta_{\min}(U) \gtrsim 1/\sqrt{k}$, the right-hand side is greater than zero and the nonconvex Slater's condition holds. We assumed for simplicity that $\rho=0$ but this is expected to hold for $\rho$ sufficiently small as well by continuity. \paragraph{\textbf{New Slater's condition}} Here we describe a variant of the Slater's condition for Program \eqref{prob:01}. \begin{definition} \textbf{(Nonconvex Slater's condition)} Let $\theta_{\min}$ be the smallest angle between to subspace and define $\psi$ to be \begin{align} \psi_{A,C} & := \begin{cases} \inf_u \, \sin \left( \theta_{\min} (\Null(A),T_C(u)) \right)\\ A u \ne 0\\ u\in \partial C \end{cases} \nonumber\\ & = \begin{cases} \inf_u \, \eta_{\min}\left(P_{T_C(u)}A^\top\right)\\ A u \ne 0\\ u\in \partial C \end{cases} \end{align} where $\partial C$ is the boundary of $C$, and $\eta_{\min}$ returns the smallest singular value. We say that Program \eqref{prob:01} satisfies the Slater's condition if $\psi_{A,C}>0$. \end{definition} As a sanity check, we have the following result. \begin{proposition}\label{prop:convex} The nonconvex Slater's condition for Program (\ref{prob:01}) implies the standard Slater's condition when $A$ is a linear operator and Program (\ref{prob:01}) is feasible. \end{proposition} \begin{proof} Suppose that the standard Slater's condition does not hold, namely, that \begin{equation} \relint(\Null(A) \cap C) = \Null(A)\cap \relint(C) = \emptyset. \label{eq:no feas} \end{equation} Since Program \eqref{prob:01} is feasible, there exists a feasible $u$, namely, $Au=0$ and $u\in C$. By \eqref{eq:no feas}, it must be that $u\in \partial C$ and that $\Null(A)$ supports $C$ at $u$ \textbf{Why?}. In particular, it follows that $\Null(A) \cap T_C(u) \ne \{0\}$ or, equivalently, $\row(A)\cap N_C(u) \ne \{0\}$. That is, there exists a unit-norm vector $v$ such that \begin{align} P_{T_C(u)}A^\top v=0, \label{eq:existence} \end{align} and consequently \begin{align} \eta_{\min}(P_{T_C(u)}A^\top) = 0. \end{align} Because $\eta_{\min}(P_{T_C(u)}A^\top)$ is a continuous function of $u$ \textbf{(Why?)}, we conclude that $\psi_{A,C}=0$, namely, the nonconvex Slater's condition also does not hold, thereby completing the proof of Proposition \ref{prop:convex}. % %Consider now the curve $g: [0,1]\rightarrow \partial C$ passing through $u$ with the direction of $A^\dagger v$, namely, %\begin{equation} %g(0) = u, %\qquad %\frac{dg}{dt}(0) = \lim_{t\rightarrow0^+}\frac{g(t)-g(0)}{t} = (A^\dagger)^\top v. %\end{equation} %\textbf{Does the above curve exist even when $C$ is not smooth?} Then note that %\begin{align} %& \eta_{\min}(P_{T_C(u)} A^\top) \nonumber\\ %& \le \lim_{t\rightarrow 0^+} \frac{\| P_{T_C(u)}A^\top Ag(t)\|}{\|Ag(t)\|} \nonumber\\ %& = \lim_{t\rightarrow 0^+} \frac{\| P_{T_C(u)}A^\top (Au + AA^\dagger v)\|}{\|Au t + A A^\dagger v + o(t)\|} \nonumber\\ %& = \lim_{t\rightarrow 0^+} \frac{\| P_{T_C(u)}A^\top ( t v +o(t))\|}{\|t v+o(t)\|} %\qquad \left( Au = 0,\,\, AA^\dagger = I \right) \nonumber\\ %& = \lim_{t\rightarrow 0^+} \frac{o(t)}{t} %\qquad \text{(see \eqref{eq:existence})} \nonumber\\ %& = 0. %\end{align} % % % \end{proof} %\paragraph{\textbf{Intuition in the convex case}} %We give some qualitative discussion about the last condition in \eqref{eq:conditions} here. Since this condition only pertains the feasibility, we consider the convex feasibility program %\begin{equation} %\begin{cases} %\min 0\\ %A u = 0\\ %u\in C. %\end{cases} %\end{equation} %Then consider the algorithm %\begin{equation} %u_{k+1} = P_C (u_k - \gamma A^T Au_ k ). %\end{equation} %To build intuition, assume that $A$ has orthonormal columns. Then, if $\gamma=1$, the algorithm above is an instance of the alternative projection algorithm. To further simplify the setup, let us assume that $C$ is the null space of a matrix$B$, namely, %\begin{equation} %\begin{cases} %\min 0\\ %A u = 0\\ %B u= 0. %\end{cases} %\end{equation} %Our algorithm simplifies to %\begin{equation} %u_{k+1} = (I - B^T B) (I - A^T A ) u_k. %\end{equation} %In this exposition, we assume that $\text{dim}(\text{null}(A))+\text{dim}(\text{null}(B))= d$ for simplicity. All these assumptions are for the sake of simplicity of our argument and can be relaxed. %The convergence is dictated by the corresponding spectral norm: %\begin{equation} %\rho = \| (I- B^TB) (I - A^TA) \| = \cos^2(\theta_1), %\end{equation} %where $\theta_1$ is the smallest angle between the subspaces $\text{null}(A)$ and $\text{null}(B)$. Then note that %\begin{equation} %\eta_{\min} ((I- B^TB) A^T) = \eta_{\min} ((I- B^TB) A^TA ) = \sin^2(\theta_1) = 1- \cos^2(\theta_1) = 1- \rho , %\end{equation} %where $\eta_{\min}$ returns the smallest singular value. The term on the far left is exactly $ \eta_{\min}$ in the condition \eqref{eq:conditions}. As $ \eta_{\min} $ reduces, the convergence rate slows down. This is also easy to visualize with two lines in the plane. \textbf{To summarize, ignoring the objective function $h$, we can think of our algorithm as alternative projections in the convex setting. There, the convergence rate is exactly determined with the $\eta_{\min}$ in \eqref{eq:conditions}.} \section{Proof of Lemma \ref{lem:bnd bnd Ak}} If $A(u_k)=b$, then \eqref{eq:bnd on Ak final} holds trivially. Otherwise, %we show that $P_{T_C(u_{k+1})} DA(u_k)^\top$ is a well-conditioned matrix, in order to lower bound \eqref{eq:bnd on Ak raw}. More specifically, for an integer $k_0$, consider a subspace \begin{align} -S _{k_0} \supseteq \bigcup_{k\ge k_0} T_{C}(u_k), +S _{K} \supseteq \bigcup_{k\in K} T_{C}(u_k), \label{eq:defn of S} \end{align} -and let $S_{k_0}$ with orthonormal columns denote a basis for this subspace, with some abuse of notation. We then assume that +and let $S_{K}$ with orthonormal columns denote a basis for this subspace, with some abuse of notation. We then assume that \begin{align} 0 < \eta_{\min} := \begin{cases} -\min_u \, \left\| S_{k_0}^\top P_{T_C(u)} ( DA(u)^\top v ) \right\| \\ +\min_u \, \left\| S_{K}^\top P_{T_C(u)} ( DA(u)^\top v ) \right\| \\ \|v\|=1\\ \|A(u)-b\|\le \rho\\ u\in C. \end{cases} \label{eq:new slater proof} \end{align} -If $ \sup_{k\ge k_0} \|A(u_k)-b\| \le \rho \in [0,\infty]$, +If $ \max_{k\in K} \|A(u_k)-b\| \le \rho $, then \eqref{eq:new slater proof} is in force and, for every $k\ge k_0$, we may write that \begin{align} & \left\| P_{T_C(u_{k+1})}( DA(u_{k})^\top (A(u_k)-b) ) \right\| \nonumber\\ & \ge \left\| P_{T_C(u_{k+1})} ( DA(u_{k+1})^\top (A(u_k)-b) ) \right\| - \left\| ( DA(u_{k+1}) - DA(u_k) )^\top (A(u_k)-b) \right\| \qquad \text{(non-expansiveness of projection)} \nonumber\\ & \ge \eta_{\min} \|A(u_k)-b\| - \left\| DA(u_{k+1}) - DA(u_k) \right\| \|A(u_k)-b\| \qquad \text{(see \eqref{eq:new slater proof})} \nonumber\\ & \ge \eta_{\min} \|A(u_k)-b\| - \lambda_A \|u_{k+1}-u_k\| \cdot \|A(u_k)-b\| \qquad \text{(see \eqref{eq:smoothness basic})} \nonumber\\ & = \left( \eta_{\min} -\lambda_A \gamma_{k} \|G_{k}\| \right) \|A(u_k)-b\| \qquad \text{(see \eqref{eq:grad map recall})} \nonumber\\ & \ge \frac{\eta_{\min}}{2} \|A(u_k)-b\|, \label{eq:well cnd} \end{align} where the last line above uses the observation that \begin{align} \lambda_A\gamma_k\|G_k\| & = \lambda_A \|u_{k+1}-u_k\| \nonumber\\ & \le \lambda_A \text{diam}(C) \nonumber\\ & \le \frac{\eta_{\min}}{2}. \qquad \text{(see \eqref{eq:cnd for well cnd})} \end{align} We can now lower bound \eqref{eq:bnd on Ak raw} by using \eqref{eq:well cnd}, namely, \begin{align} \frac{\eta_{\min}}{2\beta_{k}} \|A(u_{k})-b\|& \le \frac{1}{\beta_{k}}\| P_{T_C(u_{k+1})} ( DA(u_{k})^\top (A(u_{k})-b) )\| \qquad \text{(see \eqref{eq:well cnd})} \nonumber\\ & \le \lambda_h' + \eta_{\max} \|y_{k-1}\|+ \|G_{k}\|. \qquad \text{(see \eqref{eq:bnd on Ak raw})} % \nonumber\\ %& \le \frac{ \eta_{\min}\rho}{2\sigma_{k}}, % \qquad \text{(see \eqref{eq:to be met asympt})} % \end{align} %The step of the induction is proved similarly by replacing $k_0$ above with $k$. which completes the proof of Lemma \ref{lem:bnd bnd Ak}. \end{document} -% end of file template.tex - - Set - $X = \menge{u}{Lu=b} \cap C$. - Suppose that there exist a neighborhood $B(\overline{u};\epsilon)$ of $\overline{u}$ and - a neighborhood $B(\overline{y};\epsilon)$ of $\overline{y}$, and positive number $\alpha_1,\alpha_2$ and non-negative $\rho_1,\rho_2$ - such that $ (\forall u \in X \cap B(\overline{u};\epsilon))(\forall y \in B(\overline{y};\epsilon))$, - \begin{equation}\label{e:maic} -\scal{\nabla F_{\beta}(u,y)- \nabla F_{\beta}(\overline{u},\overline{y}) }{u-\overline{u}} - \geq \rho_1\|u-\overline{u}\|^{\alpha_1} +\rho_2\| y- \overline{y}\|^{\alpha_2}. - \end{equation} - \begin{theorem} Suppose that - $\gamma_k \geq \underline{\gamma} > 0$ and -all the conditions in Theorem \ref{t:1} is satisfied. Furthermore, - Suppose that for some $k_0$, $(u_{k})_{k\geq k_0} \subset B(\overline{u};\epsilon)$ and $(y_{k})_{k\geq k_0} \subset B(\overline{y};\epsilon)$, and - \eqref{e:maic} is satisfied for every $\beta\in (\beta_k)_{k\geq k_0}$. If $\rho_1 > 0$ or $\rho_2 > 0$, then $u_{k}\to \overline{u}$ or $y_k \to \overline{y}$. - \end{theorem} - \begin{proof} Without lost of generality, $k_0=0$. - It follows from Theorem \ref{t:1} that $\gamma_k\|(u_{k+1}-u_k)/\gamma_k\|^2 \to 0$. Since $(\gamma_k)_{k\in\NN}$ - is bounded below by $\underline{\gamma}$, we obtain $G_{\beta_k,\gamma_k}(u_k)\to 0$. - Since $\beta_{k+1}^{-1}\|Lu_{k+1}-b\|^2 \to 0$ and $(\beta_{k})_{k\in\NN}$ is bounded above by $\overline{\beta}=c$, - we get $\|Lu_{k+1}-b\|^2\to 0$. Since $L\overline{u} =b$, we can rewrite \eqref{e:fr1} as $-\nabla F_{\beta}(\overline{u},\overline{y}) \in N_{C}(\overline{u})$. Hence - \begin{equation}\label{e:aa1s} - (\forall u\in C)(\forall k\in\NN)\; \scal{\nabla F_{\beta_{k}}(\overline{u},\overline{y}) }{u-\overline{u}}\geq 0. - \end{equation} - Now, using the updating of $u_{k}$, we have $G_{\beta_k,\gamma_k}(u_k) -\nabla F_{\beta_k}(u_k,y_k) \in N_{C}(u_{k+1})$. - Hence, - \begin{equation}\label{e:aa2s} - (\forall u\in C)(\forall k\in\NN)\; \scal{-G_{\beta_k,\gamma_k}(u_k)+\nabla F_{\beta_{k}}(u_k,y_k) }{u-u_{k+1}}\geq 0. - \end{equation} - Now, we imply from \eqref{e:aa1s} and \eqref{e:aa2s} that - \begin{alignat}{2} - \label{e:las1} - \scal{-G_{\beta_k,\gamma_k}(u_k)}{\overline{u}-u_{k+1}} + \scal{\nabla F_{\beta_{k}}(u_k,y_k) -\nabla F_{\beta_{k}}(\overline{u},\overline{y}}{\overline{u}-u_{k+1}}\geq 0. - \end{alignat} - - Using the condition \eqref{e:maic}, we derive from \eqref{e:las1} that - \begin{alignat}{2} - \rho_1\|u_{n+1}-\overline{u}\|^{\alpha_1} &+\rho_2 \| y_n- \overline{y}\|^{\alpha_2}) \leq \scal{-G_{\beta_k,\gamma_k}(u_k)}{\overline{u}-u_{k+1}}\notag\\ - &\quad +\scal{\nabla F_{\beta_{k}}(u_k,y_k) -\nabla F_{\beta_{k}}(\overline{u},\overline{y})}{u_{k}-u_{k+1}} \notag\\ - & \leq (\epsilon+\gamma_k \|\nabla F_{\beta_{k}}(u_k,y_k) -\nabla F_{\beta_{k}}(\overline{u},\overline{y})\|) \| G_{\beta_k,\gamma_k}(u_k)\|\notag\\ - &\leq (2\epsilon + \|\nabla F_{\beta_{k}}(\overline{u},y_k)-\nabla F_{\beta_{k}}(\overline{u},\overline{y}) \|)\| G_{\beta_k,\gamma_k}(u_k)\|\notag\\ - &\leq (2\epsilon + \|\nabla L(\overline{u})\|\epsilon \| G_{\beta_k,\gamma_k}(u_k)\|\notag\\ -& \to 0. - \end{alignat} - Therefore, $u_{n}\to \overline{u}$ or $y_n\to \overline{y}$ provided that $\rho_1> 0$ or $\rho_2 >0$. - \end{proof} - -