Homec4science
Diffusion Nonlinear (master) Browse

R7172/Journal/journal_main.aux
bf10b3badf11master

Last Change
Raw File

journal_main.aux
View Options

\relax
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\citation{khot2011grothendieck,lovasz2003semidefinite,zhao1998semidefinite}
\citation{mossel2015consistency,song2007dependence}
\citation{singer2011angular,singer2011three}
\citation{raghavendra2008optimal}
\citation{yurtsever2018conditional}
\citation{burer2003nonlinear,burer2005local}
\citation{pataki1998rank,barvinok1995problems}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{2}{section.1}}
\newlabel{intro}{{1}{2}{Introduction}{section.1}{}}
\newlabel{prob:01}{{1}{2}{Introduction}{equation.1.1}{}}
\newlabel{eq:smoothness basic}{{2}{2}{Introduction}{equation.1.2}{}}
\@writefile{toc}{\contentsline {paragraph}{\textbf {Vignette: Burer-Monteiro splitting.}}{2}{section*.1}}
\newlabel{eq:sdp}{{3}{2}{\textbf {Vignette: Burer-Monteiro splitting.}}{equation.1.3}{}}
\citation{boumal2016non,waldspurger2018rank}
\citation{luenberger1984linear}
\citation{bertsekas2014constrained,fernandez2012local,birgin2016evaluation,cartis2018optimality,cartis2011evaluation,bolte2018nonconvex,bolte2017error,clason2018acceleration,wang2015global,wang2018convergence,jiang2019structured,guo2017convergence,yang2017alternating,li2015global}
\citation{burer2003nonlinear,burer2005local,bhojanapalli2016dropping,park2016provable,boumal2014manopt,boumal2016global,boumal2016non}
\citation{liu2019linearized}
\citation{bot2018proximal,hong2016convergence}
\citation{parikh2014proximal}
\newlabel{prob:nc}{{4}{3}{\textbf {Vignette: Burer-Monteiro splitting.}}{equation.1.4}{}}
\newlabel{eq:minmax}{{5}{3}{\textbf {Vignette: Burer-Monteiro splitting.}}{equation.1.5}{}}
\newlabel{eq:Lagrangian}{{6}{3}{\textbf {Vignette: Burer-Monteiro splitting.}}{equation.1.6}{}}
\newlabel{e:exac}{{7}{3}{\textbf {Vignette: Burer-Monteiro splitting.}}{equation.1.7}{}}
\newlabel{eq:dual-update}{{8}{3}{\textbf {Vignette: Burer-Monteiro splitting.}}{equation.1.8}{}}
\citation{bot2018proximal}
\citation{cartis2018optimality}
\citation{karimi2016linear}
\citation{bertsekas2014constrained}
\citation{bolte2018nonconvex,xu2017globally,rockafellar1993lagrange,flores2012complete}
\@writefile{toc}{\contentsline {paragraph}{\emph {\textbf {Contributions.}} }{4}{section*.2}}
\newlabel{eq:new update}{{9}{4}{\emph {\textbf {Contributions.}}}{equation.1.9}{}}
\newlabel{sec:preliminaries}{{2}{4}{Preliminaries}{section.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Preliminaries }{4}{section.2}}
\@writefile{toc}{\contentsline {paragraph}{\textbf {\emph {Notation.}}}{4}{section*.3}}
\newlabel{eq:defn of prox}{{10}{4}{\textbf {\emph {Notation.}}}{equation.2.10}{}}
\newlabel{sec:opt cnds}{{2}{5}{\textbf {\emph {{Optimality conditions.}}}}{section*.4}{}}
\@writefile{toc}{\contentsline {paragraph}{\textbf {\emph {{Optimality conditions.}}} }{5}{section*.4}}
\newlabel{e:inclu1}{{12}{5}{\textbf {\emph {{Optimality conditions.}}}}{equation.2.12}{}}
\newlabel{e:inclu2}{{13}{5}{\textbf {\emph {{Optimality conditions.}}}}{equation.2.13}{}}
\newlabel{eq:local-min}{{14}{5}{\textbf {\emph {{Optimality conditions.}}}}{equation.2.14}{}}
\@writefile{toc}{\contentsline {paragraph}{\emph {\textbf {Technical lemmas.}}}{5}{section*.5}}
\newlabel{lem:smoothness}{{1}{5}{\emph {\textbf {Technical lemmas.}}}{lemma.1}{}}
\newlabel{eq:smooth-lagrange-defn}{{15}{5}{\emph {\textbf {Technical lemmas.}}}{equation.2.15}{}}
\newlabel{eq:smoothness of Lagrangian}{{17}{5}{\emph {\textbf {Technical lemmas.}}}{equation.2.17}{}}
\citation{bolte2014proximal}
\citation{boyd2004convex}
\newlabel{def:grad map}{{1}{6}{\emph {\textbf {Technical lemmas.}}}{definition.1}{}}
\newlabel{eq:grad map}{{19}{6}{\emph {\textbf {Technical lemmas.}}}{equation.2.19}{}}
\newlabel{lem:11}{{2}{6}{\emph {\textbf {Technical lemmas.}}}{lemma.2}{}}
\newlabel{e:deslem}{{22}{6}{\emph {\textbf {Technical lemmas.}}}{equation.2.22}{}}
\newlabel{lem:eval Lipsc cte}{{3}{6}{\emph {\textbf {Technical lemmas.}}}{lemma.3}{}}
\newlabel{eq:defn of x+gamma}{{23}{6}{\emph {\textbf {Technical lemmas.}}}{equation.2.23}{}}
\newlabel{eq:defn of gamma line search}{{24}{6}{\emph {\textbf {Technical lemmas.}}}{equation.2.24}{}}
\newlabel{eq:moreover}{{25}{6}{\emph {\textbf {Technical lemmas.}}}{equation.2.25}{}}
\citation{sahin2019inexact}
\newlabel{sec:slater}{{3}{7}{Geometric Regularity}{section.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Geometric Regularity }{7}{section.3}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Solving (1) can be difficult even when it is a convex program. As an example, this figure shows a pathological case described in Section\nobreakspace {}\ref {sec:slater}, where the Slater\IeC {\textquoteright }s condition does is not met. Here, for example, the alternating projection algorithm to solve \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {prob:01}\unskip \@@italiccorr )}} converges at the rate of $O(1/\sqrt {k})$, in contrast to the (much faster) eventual linear convergence if the Slater's condition was met, namely, if $\text {null}(A)$ intersected the interior of $C$.}}{7}{figure.1}}
\newlabel{fig:convex_slater}{{1}{7}{Solving (1) can be difficult even when it is a convex program. As an example, this figure shows a pathological case described in Section~\ref {sec:slater}, where the Slater’s condition does is not met. Here, for example, the alternating projection algorithm to solve \eqref {prob:01} converges at the rate of $O(1/\sqrt {k})$, in contrast to the (much faster) eventual linear convergence if the Slater's condition was met, namely, if $\text {null}(A)$ intersected the interior of $C$}{figure.1}{}}
\newlabel{defn:nonconvex slater}{{2}{7}{Geometric Regularity}{definition.2}{}}
\newlabel{eq:new slater defn}{{26}{7}{Geometric Regularity}{equation.3.26}{}}
\citation{moreau1962decomposition}
\citation{rockafellar2015convex}
\@writefile{toc}{\contentsline {paragraph}{\textbf {\emph {Jacobian $\operatorname {D}A$.}}}{8}{section*.6}}
\@writefile{toc}{\contentsline {paragraph}{\textbf {\emph {Subspace $S$.}}}{8}{section*.7}}
\newlabel{eq:defn_nu_A}{{27}{8}{\textbf {\emph {Subspace $S$.}}}{equation.3.27}{}}
\@writefile{toc}{\contentsline {paragraph}{\emph {\textbf {Convex case.}}}{8}{section*.8}}
\newlabel{eq:nu for cvx}{{28}{8}{\emph {\textbf {Convex case.}}}{equation.3.28}{}}
\citation{karimi2016linear}
\citation{xu2017globally}
\citation{bolte2018nonconvex}
\citation{bertsekas2014constrained}
\citation{rockafellar1993lagrange}
\citation{bolte2018nonconvex}
\newlabel{prop:convex}{{1}{9}{\emph {\textbf {Convex case.}}}{proposition.1}{}}
\@writefile{toc}{\contentsline {paragraph}{\emph {\textbf {Beyond the Slater's condition.}}}{9}{section*.9}}
\@writefile{toc}{\contentsline {paragraph}{\emph {\textbf {Relation to prior work.}}}{9}{section*.10}}
\citation{bertsekas1976penalty}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces This figure shows that geometric regularity in Definition\nobreakspace {}\ref {defn:nonconvex slater} is often a local property. More specifically, in problem\nobreakspace {}\textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {prob:01}\unskip \@@italiccorr )}}, suppose that $f\equiv 0$, $g=1_C$ is the indicator function on the disk $C$, and $A$ is a linear operator with the null space depicted in the figure. Then the objective in\nobreakspace {}\textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {eq:new slater defn}\unskip \@@italiccorr )}}, evaluated at the red point, is zero. To ensure that the geometric regularity holds ($nu(1_C,A,\mathbb {R}^d,\rho ,\rho ')>0$), we must choose a small enough $\rho $. That is, the geometric regularity holds if the feasibility gap is sufficiently small. {\color {magenta} \textbf {AE: add the figure!}}}}{10}{figure.2}}
\newlabel{fig:counter}{{2}{10}{This figure shows that geometric regularity in Definition~\ref {defn:nonconvex slater} is often a local property. More specifically, in problem~\eqref {prob:01}, suppose that $f\equiv 0$, $g=1_C$ is the indicator function on the disk $C$, and $A$ is a linear operator with the null space depicted in the figure. Then the objective in~\eqref {eq:new slater defn}, evaluated at the red point, is zero. To ensure that the geometric regularity holds ($nu(1_C,A,\mathbb {R}^d,\rho ,\rho ')>0$), we must choose a small enough $\rho $. That is, the geometric regularity holds if the feasibility gap is sufficiently small. \notea {add the figure!}}{figure.2}{}}
\newlabel{sec:holal}{{4}{10}{Homotopy Linearized AL Algorithm}{section.4}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Homotopy Linearized AL Algorithm }{10}{section.4}}
\newlabel{lem:bnd bnd Ak}{{4}{10}{Homotopy Linearized AL Algorithm}{lemma.4}{}}
\newlabel{eq:low-bnd-on-nu-lemma}{{29}{10}{Homotopy Linearized AL Algorithm}{equation.4.29}{}}
\newlabel{Algo:2}{{4}{11}{Homotopy Linearized AL Algorithm}{section.4}{}}
\@writefile{loa}{\contentsline {algorithm}{\numberline {1}{\ignorespaces HoLAL algorithm for solving problem \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {prob:01}\unskip \@@italiccorr )}}}}{11}{algorithm.1}}
\newlabel{eq:bnd on Ak final}{{30}{11}{Homotopy Linearized AL Algorithm}{equation.4.30}{}}
\newlabel{thm:main}{{1}{11}{Homotopy Linearized AL Algorithm}{theorem.4.1}{}}
\citation{xu2017accelerated}
\citation{bot2018proximal}
\citation{sahin2019inexact,cartis2018optimality}
\newlabel{eq:low-bnd-on-regularity}{{33}{12}{Homotopy Linearized AL Algorithm}{equation.4.33}{}}
\newlabel{eq:rates-thm}{{34}{12}{Homotopy Linearized AL Algorithm}{equation.4.34}{}}
\@writefile{toc}{\contentsline {paragraph}{\textbf {\emph {Convergence rates.}}}{12}{section*.11}}
\newlabel{eq:rates-summary}{{35}{12}{\textbf {\emph {Convergence rates.}}}{equation.4.35}{}}
\citation{gomez2019fast}
\citation{Glowinski1975,Gabay1976,Boyd2011}
\@writefile{toc}{\contentsline {paragraph}{\emph {\textbf {Geometric regularity.}}}{13}{section*.12}}
\@writefile{toc}{\contentsline {paragraph}{\emph {\textbf {Smoothness.}}}{13}{section*.13}}
\@writefile{toc}{\contentsline {paragraph}{\emph {\textbf {Subspace $S$.}}}{13}{section*.14}}
\@writefile{toc}{\contentsline {paragraph}{\textbf {\emph {Faster rates.}}}{13}{section*.15}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Linearized Alternating Direction Method of Multipliers}{13}{section.5}}
\newlabel{eq:admm 2}{{36}{13}{Linearized Alternating Direction Method of Multipliers}{equation.5.36}{}}
\newlabel{eq:lagrangian admm}{{37}{14}{Linearized Alternating Direction Method of Multipliers}{equation.5.37}{}}
\newlabel{eq:admm main}{{38}{14}{Linearized Alternating Direction Method of Multipliers}{equation.5.38}{}}
\newlabel{eq:grad map admm}{{39}{14}{Linearized Alternating Direction Method of Multipliers}{equation.5.39}{}}
\newlabel{eq:line search x admm}{{42}{14}{Linearized Alternating Direction Method of Multipliers}{equation.5.42}{}}
\newlabel{eq:defn of gamma line search admm}{{43}{14}{Linearized Alternating Direction Method of Multipliers}{equation.5.43}{}}
\newlabel{eq:smoothness basics admm}{{45}{15}{Linearized Alternating Direction Method of Multipliers}{equation.5.45}{}}
\newlabel{Algo:3}{{5}{15}{Linearized Alternating Direction Method of Multipliers}{equation.5.45}{}}
\@writefile{loa}{\contentsline {algorithm}{\numberline {2}{\ignorespaces Linearized ADMM for solving problem \textup {\hbox {\mathsurround \z@ \normalfont (\ignorespaces \ref {eq:admm main}\unskip \@@italiccorr )}}}}{15}{algorithm.2}}
\newlabel{thm:main admm}{{2}{15}{Linearized Alternating Direction Method of Multipliers}{theorem.5.2}{}}
\newlabel{eq:low-bnd-on-regularity}{{47}{16}{Linearized Alternating Direction Method of Multipliers}{equation.5.47}{}}
\newlabel{eq:rates-thm}{{48}{16}{Linearized Alternating Direction Method of Multipliers}{equation.5.48}{}}
\newlabel{sec:experiments}{{6}{16}{Experiments}{section.6}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6}Experiments }{16}{section.6}}
\newlabel{sec:proof-of-main}{{A}{16}{Proof of Theorem \ref {thm:main}}{appendix.A}{}}
\@writefile{toc}{\contentsline {section}{\numberline {A}Proof of Theorem \ref {thm:main} }{16}{appendix.A}}
\newlabel{eq:update uk recall}{{49}{16}{Proof of Theorem \ref {thm:main}}{equation.A.49}{}}
\newlabel{eq:y update recall}{{50}{16}{Proof of Theorem \ref {thm:main}}{equation.A.50}{}}
\newlabel{eq:grad map recall}{{51}{16}{Proof of Theorem \ref {thm:main}}{equation.A.51}{}}
\newlabel{eq:smoothness rep}{{53}{17}{Proof of Theorem \ref {thm:main}}{equation.A.53}{}}
\newlabel{eq:y update unfolded}{{54}{17}{Proof of Theorem \ref {thm:main}}{equation.A.54}{}}
\newlabel{eq:key ineq}{{55}{17}{Proof of Theorem \ref {thm:main}}{equation.A.55}{}}
\newlabel{eq:long chain broken}{{57}{18}{Proof of Theorem \ref {thm:main}}{equation.A.57}{}}
\newlabel{eq:defn mu}{{58}{18}{Proof of Theorem \ref {thm:main}}{equation.A.58}{}}
\newlabel{eq:beta n sigma}{{59}{18}{Proof of Theorem \ref {thm:main}}{equation.A.59}{}}
\newlabel{eq:consequences}{{60}{19}{Proof of Theorem \ref {thm:main}}{equation.A.60}{}}
\newlabel{eq:long chain broken 10}{{61}{19}{Proof of Theorem \ref {thm:main}}{equation.A.61}{}}
\newlabel{eq:to be used for feas pre}{{63}{20}{Proof of Theorem \ref {thm:main}}{equation.A.63}{}}
\newlabel{eq:defn of BK}{{65}{20}{Proof of Theorem \ref {thm:main}}{equation.A.65}{}}
\newlabel{eq:to be used for feas}{{66}{20}{Proof of Theorem \ref {thm:main}}{equation.A.66}{}}
\newlabel{eq:to be used later on}{{67}{20}{Proof of Theorem \ref {thm:main}}{equation.A.67}{}}
\newlabel{eq:final bnd on G}{{68}{20}{Proof of Theorem \ref {thm:main}}{equation.A.68}{}}
\newlabel{eq:final bnd on feas gap}{{69}{21}{Proof of Theorem \ref {thm:main}}{equation.A.69}{}}
\newlabel{eq:overall bnd raw}{{70}{21}{Proof of Theorem \ref {thm:main}}{equation.A.70}{}}
\newlabel{eq:dual growth}{{71}{21}{Proof of Theorem \ref {thm:main}}{equation.A.71}{}}
\newlabel{eq:Bk evaluated}{{72}{21}{Proof of Theorem \ref {thm:main}}{equation.A.72}{}}
\newlabel{eq:smoothness at k}{{73}{21}{Proof of Theorem \ref {thm:main}}{equation.A.73}{}}
\newlabel{eq:low bnd on gammas}{{74}{21}{Proof of Theorem \ref {thm:main}}{equation.A.74}{}}
\newlabel{eq:exact dependences}{{76}{22}{Proof of Theorem \ref {thm:main}}{equation.A.76}{}}
\newlabel{eq:rate of feas gap}{{77}{22}{Proof of Theorem \ref {thm:main}}{equation.A.77}{}}
\newlabel{eq:suff cnd on nu-1}{{79}{22}{Proof of Theorem \ref {thm:main}}{equation.A.79}{}}
\newlabel{eq:cnd-on-nu-proof}{{82}{23}{Proof of Theorem \ref {thm:main}}{equation.A.82}{}}
\newlabel{sec:ADMM theory}{{B}{23}{Proof of Theorem \ref {thm:main admm}}{appendix.B}{}}
\@writefile{toc}{\contentsline {section}{\numberline {B}Proof of Theorem \ref {thm:main admm} }{23}{appendix.B}}
\newlabel{lem:smoothness admm}{{5}{23}{Proof of Theorem \ref {thm:main admm}}{lemma.5}{}}
\newlabel{eq:smoothness of Lagrangian admm}{{85}{23}{Proof of Theorem \ref {thm:main admm}}{equation.B.85}{}}
\newlabel{eq:local-lipschitz-admm}{{86}{23}{Proof of Theorem \ref {thm:main admm}}{equation.B.86}{}}
\newlabel{def:grad map admm}{{3}{23}{Proof of Theorem \ref {thm:main admm}}{definition.3}{}}
\newlabel{eq:grad map admm}{{87}{23}{Proof of Theorem \ref {thm:main admm}}{equation.B.87}{}}
\newlabel{lem:11 admm}{{6}{23}{Proof of Theorem \ref {thm:main admm}}{lemma.6}{}}
\newlabel{e:deslem}{{6}{23}{Proof of Theorem \ref {thm:main admm}}{equation.B.88}{}}
\newlabel{eq:deslem admm}{{89}{23}{Proof of Theorem \ref {thm:main admm}}{equation.B.89}{}}
\newlabel{lem:eval Lipsc cte admm}{{7}{24}{Proof of Theorem \ref {thm:main admm}}{lemma.7}{}}
\newlabel{eq:defn of gamma line search admm}{{91}{24}{Proof of Theorem \ref {thm:main admm}}{equation.B.91}{}}
\newlabel{eq:moreover admm}{{92}{24}{Proof of Theorem \ref {thm:main admm}}{equation.B.92}{}}
\newlabel{eq:ADMM updates recall}{{93}{24}{Proof of Theorem \ref {thm:main admm}}{equation.B.93}{}}
\newlabel{eq:decrease twice}{{96}{24}{Proof of Theorem \ref {thm:main admm}}{equation.B.96}{}}
\newlabel{eq:defn of w}{{97}{25}{Proof of Theorem \ref {thm:main admm}}{equation.B.97}{}}
\newlabel{eq:to-be-projected-admm}{{103}{25}{Proof of Theorem \ref {thm:main admm}}{equation.B.103}{}}
\newlabel{sec:proof of smoothness lemma}{{C}{25}{Proof of Lemma \ref {lem:smoothness}}{appendix.C}{}}
\@writefile{toc}{\contentsline {section}{\numberline {C}Proof of Lemma \ref {lem:smoothness}}{25}{appendix.C}}
\newlabel{sec:proof of descent lemma}{{D}{26}{Proof of Lemma \ref {lem:11}}{appendix.D}{}}
\@writefile{toc}{\contentsline {section}{\numberline {D}Proof of Lemma \ref {lem:11}}{26}{appendix.D}}
\newlabel{eq:descent pr 1}{{111}{26}{Proof of Lemma \ref {lem:11}}{equation.D.111}{}}
\newlabel{eq:opt of prox}{{112}{26}{Proof of Lemma \ref {lem:11}}{equation.D.112}{}}
\citation{parikh2014proximal}
\newlabel{sec:proof of linesearch}{{E}{27}{Proof of Lemma \ref {lem:eval Lipsc cte}}{appendix.E}{}}
\@writefile{toc}{\contentsline {section}{\numberline {E}Proof of Lemma \ref {lem:eval Lipsc cte}}{27}{appendix.E}}
\newlabel{eq:optimality of uplus}{{114}{27}{Proof of Lemma \ref {lem:eval Lipsc cte}}{equation.E.114}{}}
\newlabel{sec:proof of bnd Ak}{{F}{27}{Proof of Lemma \ref {lem:bnd bnd Ak}}{appendix.F}{}}
\@writefile{toc}{\contentsline {section}{\numberline {F}Proof of Lemma \ref {lem:bnd bnd Ak} }{27}{appendix.F}}
\newlabel{eq:bndness in slater proof}{{116}{27}{Proof of Lemma \ref {lem:bnd bnd Ak}}{equation.F.116}{}}
\newlabel{eq:to be projected}{{118}{27}{Proof of Lemma \ref {lem:bnd bnd Ak}}{equation.F.118}{}}
\newlabel{eq:defn of dual cone}{{119}{27}{Proof of Lemma \ref {lem:bnd bnd Ak}}{equation.F.119}{}}
\newlabel{eq:before Sk}{{120}{27}{Proof of Lemma \ref {lem:bnd bnd Ak}}{equation.F.120}{}}
\newlabel{eq:defn of S}{{121}{27}{Proof of Lemma \ref {lem:bnd bnd Ak}}{equation.F.121}{}}
\newlabel{eq:pre norm}{{122}{28}{Proof of Lemma \ref {lem:bnd bnd Ak}}{equation.F.122}{}}
\newlabel{eq:post norm}{{123}{28}{Proof of Lemma \ref {lem:bnd bnd Ak}}{equation.F.123}{}}
\newlabel{eq:slater proof 0}{{124}{28}{Proof of Lemma \ref {lem:bnd bnd Ak}}{equation.F.124}{}}
\newlabel{eq:slater proof 1}{{126}{28}{Proof of Lemma \ref {lem:bnd bnd Ak}}{equation.F.126}{}}
\bibstyle{plain}
\bibdata{bibliography.bib}
\bibcite{barvinok1995problems}{1}
\bibcite{bertsekas1976penalty}{2}
\bibcite{bertsekas2014constrained}{3}
\bibcite{bhojanapalli2016dropping}{4}
\bibcite{birgin2016evaluation}{5}
\newlabel{sec:proof of prop}{{G}{29}{Proof of Proposition \ref {prop:convex}}{appendix.G}{}}
\@writefile{toc}{\contentsline {section}{\numberline {G}Proof of Proposition \ref {prop:convex} }{29}{appendix.G}}
\newlabel{eq:no feas}{{131}{29}{Proof of Proposition \ref {prop:convex}}{equation.G.131}{}}
\newlabel{eq:counter-assump}{{132}{29}{Proof of Proposition \ref {prop:convex}}{equation.G.132}{}}
\newlabel{eq:counter-assump-rewritten}{{133}{29}{Proof of Proposition \ref {prop:convex}}{equation.G.133}{}}
\bibcite{bolte2017error}{6}
\bibcite{bolte2018nonconvex}{7}
\bibcite{bot2018proximal}{8}
\bibcite{boumal2016global}{9}
\bibcite{boumal2014manopt}{10}
\bibcite{boumal2016non}{11}
\bibcite{boyd2004convex}{12}
\bibcite{burer2003nonlinear}{13}
\bibcite{burer2005local}{14}
\bibcite{cartis2018optimality}{15}
\bibcite{cartis2011evaluation}{16}
\bibcite{clason2018acceleration}{17}
\bibcite{fernandez2012local}{18}
\bibcite{flores2012complete}{19}
\bibcite{gomez2019fast}{20}
\bibcite{guo2017convergence}{21}
\bibcite{hong2016convergence}{22}
\bibcite{jiang2019structured}{23}
\bibcite{karimi2016linear}{24}
\bibcite{khot2011grothendieck}{25}
\bibcite{li2015global}{26}
\bibcite{liu2019linearized}{27}
\bibcite{lovasz2003semidefinite}{28}
\bibcite{moreau1962decomposition}{29}
\bibcite{mossel2015consistency}{30}
\bibcite{parikh2014proximal}{31}
\bibcite{park2016provable}{32}
\bibcite{pataki1998rank}{33}
\bibcite{raghavendra2008optimal}{34}
\bibcite{rockafellar1993lagrange}{35}
\bibcite{rockafellar2015convex}{36}
\bibcite{sahin2019inexact}{37}
\bibcite{singer2011angular}{38}
\bibcite{singer2011three}{39}
\bibcite{song2007dependence}{40}
\bibcite{waldspurger2018rank}{41}
\bibcite{wang2018convergence}{42}
\bibcite{wang2015global}{43}
\bibcite{xu2017accelerated}{44}
\bibcite{xu2017globally}{45}
\bibcite{yang2017alternating}{46}
\bibcite{yurtsever2018conditional}{47}
\bibcite{zhao1998semidefinite}{48}
c4science · Help