where $1_C$ is the convex indicator function on $C$ and $\b^*>0$. Above, $\L$ is the augmented Lagrangian of Program \eqref{eq:main}. Therefore, $(u^*,y^*)$ is a stationary point of Program \eqref{eq:main} if
\begin{align}
\begin{cases}
0\in \nabla_u \L(u^*,y^*,\b^*) + N_C(u^*) \\
L(u^*) = 0,
\end{cases}
\label{eq:opt cnd}
\end{align}
where $DL(u^*)$ is the Jacobian of $L$ and $N_C(u^*)$ is the normal cone of $C$, both at $u^*$. Note that $u^*$ is a local minimum if
where $T_C(u^*)$ is the tangent cone of $C$ at $u^*$. Alternatively, if the above second form takes both positive and negative values on $T_C(u^*)$, then $u^*$ is a strict saddle point.
%Note that $u$ is a stationary point of Program \eqref{eq:main} if
%\begin{align}
%\begin{cases}
%\delta\in T_C(u) \\
%\nabla h(u)^T \delta = 0\\
%L(u)=0
%\end{cases}
%\Longrightarrow
%DL(u)\delta = 0,
%\end{align}
%where $T_C(u)$ is the tangent cone to $C$ at $u$. Equivalently, $u$ is a stationary point of Program \eqref{eq:main} if
To solve this program, Bang et al. suggested the first-order algorithm
for positive-valued $b,c$ specified in their write-up. {\color{blue} Is the last claim correct?}
It is easy to verify that a limit point $z^* = (u^*,y^*,\b^*,\g^*)$ of Algorithm \eqref{eq:main alg} satisfies \eqref{eq:opt cnd}.
Consequently, any limit point of Algorithm \eqref{eq:main alg} is a stationary point of Program \eqref{eq:main} and vice verse. However, without limiting the choice of $h$ (to convex for example), we cannot ensure that Algorithm \eqref{eq:main alg} finds a local minimum of Program \eqref{eq:main}. Our goal is to show that, almost surely, Algorithm \eqref{eq:main alg} avoids (strict) saddle points of Program \eqref{eq:main}. For example, suppose that Program \eqref{eq:main} does not have any non-strict saddle points. Under this assumption, if Algorithm \eqref{eq:main alg} converges, then the limit point is a local minimum, almost surely.
%Then we may write the above algorithm as a dynamical system that satisfies
%\begin{align}
%z_{k+1}:=
%\l[
%\begin{array}{c}
%u_{k+1}\\
%y_{k+1}\\
%\b_{k+1}\\
%\g_{k+1}
%\end{array}
%\r]
%& =
%\l[
%\begin{array}{c}
%P_C (u_k- \g_k\nabla_u \L_k(u_k,y_k)\\
%y_k+ \frac{L(u_{k+1})}{2\b_k}\\
%b(u_k,y_k,\b_k,\g_k)\\
%c(u_k,y_k,\b_k,\g_k)
%\end{array}
%\r].
%\end{align}
\section{Analysis}
Let us use the shorthand $z=(u,y,\b,\g)$ and consider the dynamical system specified by
\begin{align}
z_{k+1}
& :=
\l[
\begin{array}{c}
P_C (u_k- \g_k\nabla_u \L(z_k))\\
y_k+ \frac{L(u_{k})}{2\b_k}\\
b(z_k)\\
c(z_k)
\end{array}
\r]
=: O
\l(z_k
\r)
.
\label{eq:sys}
\end{align}
It is easy to verify that any fixed point of the System \eqref{eq:sys} is a limit point of Algorithm \eqref{eq:main alg}. Moreover, when close enough to a stationary point (equivalently, fixed point) $z^*$, the future iterates of Algorithm \ref{eq:main alg} are attracted (dispelled) from $z^*$ if and only if the future iterates of System \eqref{eq:sys} are attracted (dispelled) from $z^*$. {\color{blue} Haven't verified the last claim.}
%with the same starting point, the sequence produced by System \eqref{eq:sys} converges to a fixed point $z^*$ if and only if the sequence produced by Algorithm \eqref{eq:main alg} converges to $z^*$.
It therefore suffices to study System \eqref{eq:sys} in what follows.
\begin{lem}
$O$ is an isomorphism in a neighborhood of $z^*$. That is, $O$ is continuous and has a continuous inverse. {\color{blue} I don't know if this claim is correct... For it to be correct, it might require $C$ to have bounded \emph{reach}. }
\end{lem}
Consider a fixed point $z^*$ of System \eqref{eq:sys} and a point $z=z^*+\d_z$ in its vicinity, where $\d_z=(\d_u,\d_y,\d_\b,\d_\g)$ and $\d_u\in T_C(u^*)$. We may then write that
To simplify the first entry of the vector above, let $P_{T_C(u^*)}$ be the orthogonal projection onto the tangent cone of $C$ at $u^*$, and note that {\color{blue} Is the first identity below correct?}
Let us now recall the stable manifold theorem, e.g., Theorem 3 in ``Behavior of accelerated gradient methods near critical points of non-convex problems''.
\begin{thm}
\textbf{\emph{(Stable manifold theorem)}} Suppose that $z^*$ is a fixed point of $O$. Suppose also that $O$ is itself a diffeomorphism in a neighborhood of $z^*$. Suppose lastly that $DO(z^*)$ has an eigenvalue outside of the unit circle. Then the local stable manifold $\S_{z^*} \ni z^*$ has measure zero. In particular, there exists a neighborhood $B\ni z^*$ such that, if $O^k(z)\in B$ for all $k$, then $z\in \S_{z^*}$. {\color{blue} I think we need a version of this theorem that holds for an isomorphism, namely, just continuous with a continuous inverse. Can we find one in the book ``Global stability of dynamical systems''? I have the book, if needed. }
\end{thm}
Consider the sequence $\{z_k\}$ generated by System \eqref{eq:sys}. Suppose that $\{z_k\}$ converges to $z^*$. Then, for sufficiently large $k$, $z_k$ belongs to the stable manifold at $z^*$. That is, $z_0\in O^{-k}(\S_{z^*})$. Since $\S_{z^*}$ has measure zero, so does $O^{-k}(\S_{z_k})$. That is, almost surely, System \eqref{eq:sys} (or equivalently Algorithm \eqref{eq:main alg}) does not converge to the strict saddle point $z^*$.