\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small {\em 
Electronic Journal of Differential Equations}, 
Vol. 2007(2007), No. 32, pp. 1--6.\newline 
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/32\hfil Continuous descent methods]
{Stability of convergent continuous descent methods}

\author[S. Aizicovici, S. Reich, A. J. Zaslavski \hfil EJDE-2007/32\hfilneg]
{Sergiu Aizicovici, Simeon Reich, Alexander J. Zaslavski}  % in alphabetical order

\address{Sergiu Aizicovici \newline
Department of Mathematics, Ohio University, Athens, OH 45701-2979, USA}
\email{aizicovi@math.ohiou.edu}

\address{Simeon Reich \newline
Department of Mathematics, The Technion-Israel Institute of
Technology, 32000 Haifa, Israel}
\email{sreich@tx.technion.ac.il}

\address{Alexander J. Zaslavski \newline
Department of Mathematics, The Technion-Israel Institute of
Technology, 32000 Haifa, Israel}
\email{ajzasl@tx.technion.ac.il}

\thanks{Submitted September 3, 2006. Published February 22, 2007.}
\subjclass[2000]{37L99, 47J35, 49M99, 54E35, 54E50, 54E52, 90C25}
\keywords{Complete uniform space; convex function;
descent method; \hfill\break\indent generic property; initial value problem}

\begin{abstract}
 We consider continuous descent methods for the
 minimization of convex functions defined on a general Banach
 space. In our previous work we showed that most of them (in the
 sense of Baire category) converged. In the present paper we show
 that convergent continuous descent methods are stable under small
 perturbations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

The study of discrete and continuous descent methods is an
important topic in optimization theory and in dynamical systems.
See, for example, \cite{a1,a2,c1,h1,n1,r1,r2,r3,r4}.
Given a continuous convex
function $f$ on a Banach space $X$, we associate with $f$ a
complete metric space of vector fields $V:X \to X$ such that
$f^0(x,Vx) \le 0$ for all $x \in X$. Here $f^0(x,u)$ is the
right-hand derivative of $f$ at $x$ in the direction of $u \in X$.
To each such vector field there correspond two gradient-like
iterative processes. In the papers \cite{r1,r2}, it is shown that for
most of these vector fields, both iterative processes generate
sequences $\{x_n\}_{n=1}^{\infty}$ such that the sequences
$\{f(x_n)\}_{n=1}^{\infty}$ tend to $\inf(f)$ as $n \to \infty$.
Here by ``most'' we mean an everywhere dense $G_{\delta}$ subset
of the space of vector fields (cf., for example, \cite{d1,r1,v1}). In
the recent paper \cite{a2}, we studied the convergence of the
trajectories of an analogous continuous dynamical system governed
by such vector fields to the point where the function $f$ attains
its infimum. The first attempt to examine such continuous descent
methods was made in \cite{r4}. However, it is assumed there that the
convex function $f$ is Lipschitz on all bounded subsets of $X$. No
such assumption was made in \cite{a2}. We remark in passing that
continuous descent methods for the minimization of Lipschitz (not
necessarily convex) functions are studied in \cite{a1}.

In the present paper, we show that convergent continuous descent
methods are stable under small perturbations (see Theorem \ref{thm1.2}
below).

More precisely,
let $(X,\|\cdot\|)$ be a Banach space and let $f:X \to R^1$ be a convex
continuous function which satisfies the following conditions:
\begin{itemize}
\item[C(i)] $\lim_{\|x\| \to \infty}f(x)=\infty$;

\item[C(ii)] there is a point $\bar x \in X$ such that $f(\bar x) \le
f(x)$ for all $x \in X$;

\item[C(iii)] if $\{x_n\}_{n=1}^{\infty} \subset X$ and $\lim_{n \to \infty}
f(x_n)=f(\bar x)$, then $\lim_{n \to \infty}x_n=\bar x.$

\end{itemize}
By C(iii), the point $\bar x$, where the minimum of $f$ is attained,
is unique.

For each $x \in X$, let
\begin{equation}
f^0(x,u)=\lim_{t \to 0^+}[f(x+tu)-f(x)]/t,\quad u \in X. \label{e1.1}
\end{equation}
For each $x \in X$ and $r>0$, set
\begin{equation}
B(x,r)=\{z \in X:\;\|z-x\|\le r\} ,\quad  B(r)=B(0,r). \label{e1.2}
\end{equation}
For each mapping $A:X \to X$ and each $r>0$, put
\begin{equation}
\text{Lip}(A,r)=\sup\{\|Ax-Ay\|/\|x-y\|:\; x,y \in B(r),
\text{ and } x \not =y\}. \label{e1.3}
\end{equation}
Denote by $\mathcal{A}_l$ the set of all mappings
$V:X \to X$ such that $\text{Lip}(V,r)<\infty$ for each positive
$r$ (this means that the restriction of $V$ to any bounded subset
of $X$ is Lipschitz) and $f^0(x,Vx) \le 0$ for all $x \in X$.

For the set $\mathcal{A}_l$ we consider the uniformity determined
by the base
\begin{equation}
\begin{aligned}
E_s(n,\epsilon)=\big\{&(V_1,V_2) \in \mathcal{A}_l\times \mathcal{A}_l:\;
\text{Lip}(V_1-V_2,n) \le \epsilon \\
&\text{and } \|V_1x-V_2 x\| \le \epsilon \text { for all } x \in
B(n)\big\}.
\end{aligned}\label{e1.4}
\end{equation}
Clearly, this uniform space $\mathcal{A}_l$ is metrizable and  complete. The
topology induced by this uniformity in $\mathcal{A}_l$ will be called
the strong topology.

We will also equip the space $\mathcal{A}_l$ with the uniformity determined by
the base
\begin{equation}
E_w(n,\epsilon)=\{(V_1,V_2) \in \mathcal{A}_l \times \mathcal{A}_l:\;
\|V_1x-V_2x\| \le \epsilon
\text { for all } x \in B(n)\} \label{e1.5}
\end{equation}
where $n,\epsilon >0$. The topology induced by this uniformity will be
called the weak topology.

The following existence result was proved in \cite[Section 3]{a2}.

\begin{proposition} \label{prop1.1}
Let $x_0 \in X$ and $V \in \mathcal{A}_l$. Then there
exists a unique continuously differentiable mapping
$x:[0,\infty) \to X$ such that
\begin{gather*}
x'(t)=Vx(t),\quad t \in [0,\infty),\\
x(0)=x_0.
\end{gather*}
\end{proposition}

We now recall the main result of \cite{a2}.

\begin{theorem} \label{thm1.1}
There exists a set $\mathcal{F} \subset \mathcal{A}_l$ which is a
countable intersection of open (in the weak
topology) everywhere dense (in the strong topology) subsets of
$\mathcal{A}_l$ such that for each $V \in \mathcal{F}$, the following
property holds:

For each $\epsilon >0$ and each $n>0$, there exist
$T_{\epsilon,n}>0$ and a neighborhood $\mathcal{U}$ of $V$ in
$\mathcal{A}_l$ with the
weak topology such that for each $W \in \mathcal{U}$ and each
differentiable mapping $y:[0,\infty) \to X$ satisfying
$$
|f(y(0))| \le n \quad \text{and}\quad
 y'(t)=Wy(t)\text { for all } t \ge 0,
$$
the inequality $\|y(t)-\bar x\| \le \epsilon$ holds for all
$t \ge T_{\epsilon, n}$.
\end{theorem}

Denote by $\mathcal{F}_*$ the set of all $V \in \mathcal{A}_l$ which
have the following property:
\begin{itemize}
\item[(P1)] For each $\epsilon >0$ and each $n>0$, there exists
$T_{\epsilon, n}>0$ such that for each differentiable mapping
$y:[0,\infty) \to X$ satisfying
$$|f(y(0))| \le n \quad\text{and}\quad
 y'(t)=Vy(t)\quad \text {for all } t \ge 0,
$$
the inequality $\|y(t)-\bar x\| \le \epsilon$ holds for some
$t \in [0,T_{\epsilon,n}]$.
\end{itemize}
By Theorem \ref{thm1.1}, $\mathcal{F}_*$ contains a subset which is a countable
intersection of open (in the weak topology) everywhere dense (in
the strong topology) subsets of $\mathcal{A}_l$.

We denote by $\mathcal{A}$ the set of all mappings $V:X \to X$ which
are bounded on bounded subsets of $X$ and satisfy $f^0(x,Vx) \le
0$ for all $x \in X$. Clearly, $\mathcal{A}_l \subset \mathcal{A}$.

For the set $\mathcal{A}$ we consider the uniformity determined by the base
\begin{equation}
G_w(n,\epsilon)=\{(V_1,V_2) \in \mathcal{A} \times \mathcal{A}:
\;\|V_1x-V_2x\| \le \epsilon \; \forall x \in B(n)\},\label{e1.6}
\end{equation}
 where $n,\epsilon >0$.
Clearly, the space $\mathcal{A}$ with this uniformity is metrizable. We
are now ready to formulate the main result of the present paper.

\begin{theorem} \label{thm1.2}
Let $V \in \mathcal{F}_*$ and $n,\epsilon>0$.
Then there exist $T_{\epsilon, n}>0$ and a neighborhood $\mathcal{U}$
of $V$ in $\mathcal{A}$  such that for each $W \in \mathcal{U}$, each
$T \ge T_{\epsilon,n}$ and each $x \in W^{1,1}(0,T;X)$ satisfying
\begin{equation}
x'(t)=Wx(t),\quad t \in [0,T] \text { a.e.}, \label{e1.7}
\end{equation}
and
\begin{equation}
|f(x(0))| \le n, \label{e1.8}
\end{equation}
the inequality $\|x(t)-\bar x\| \le \epsilon$ holds for all $t \in
[T_{\epsilon, n},T]$.
\end{theorem}

Our paper is organized as follows. An auxiliary result,
Proposition \ref{prop2.1}, is presented in Section 2. Our main result,
Theorem \ref{thm1.2}, is proved in Section 3.

\section{An auxiliary result}

Let the mapping $x$ belong to the Sobolev space $W^{1,1}(0,T;X)$,
i.e. (see, e.g., \cite{b1}),
$$
x(t)=x_0+\int_0^tu(s)ds,\; t \in [0,T],
$$
where $T>0$, $x_0 \in X$ and $u \in L^1(0,T;X)$. Then $x:[0,T] \to
X$ is absolutely continuous and $x'(t)=u(t)$ for a.e. $t \in
[0,T]$. Recall that the function $f:X \to R^1$ is assumed to be
convex and continuous and therefore it is, in fact, locally
Lipschitz. It follows that its restriction to the set $\{x(t):\; t
\in [0,T]\}$ is Lipschitz. Indeed, since this set is compact, the
restriction of $f$ to it is Lipschitz.

Hence the function
$(f\circ x)(t):=f(x(t))$, $t \in [0,T]$, is absolutely continuous.
It follows that for almost every $t \in [0,T]$, both the derivatives
$x'(t)$ and $(f\circ x)'(t)$ exist:
\begin{gather*}
x'(t)=\lim_{h \to 0}h^{-1}[x(t+h)-x(t)],\\
(f\circ x)'(t)=\lim_{h \to 0}h^{-1}[f(x(t+h))-f(x(t))].
\end{gather*}

We now quote \cite[Proposition 3.1]{r4}.

\begin{proposition} \label{prop2.1}
Assume that $t \in [0,T]$ and that both the derivatives $x'(t)$
and $(f\circ x)'(t)$ exist. Then
$$
(f\circ x)'(t)=\lim_{h \to 0}h^{-1}[f(x(t)+hx'(t))-f(x(t))].
$$
\end{proposition}

\section{Proof of Theorem \ref{thm1.2}}

By C(i), there is $n_1 >n$ such that
\begin{equation}
\text { if } z \in X \text { and } f(z)\le n,\;
\text { then } \|z\| \le n_1.
\label{e3.1}
\end{equation}
By C(iii), there is $\delta_1>0$ such that
\begin{equation}
\text { if } z \in X \text { and } f(z) \le f(\bar x)+\delta_1, \text {
then } \|z-\bar x\| \le \epsilon. \label{e3.2}
\end{equation}
Since $f$ is continuous, there is $\epsilon_1 >0$ such that
\begin{equation}
|f(\bar x)-f(z)| \le \delta_1 \text { for each } z \in X \text {
satisfying } \|z-\bar x\| \le \epsilon_1. \label{e3.3}
\end{equation}
Since $V \in \mathcal{A}_l$, there is $L>1$ such that
\begin{equation}
\|Vz_1-Vz_2\|\le L\|z_1-z_2\| \text { for all }
z_1,z_2 \in B(n_1). \label{e3.4}
\end{equation}
By (P1), there is $T_0>0$ such that
the following property holds:
\begin{itemize}
\item[(P2)]
For each differentiable mapping $y:[0,\infty)
\to X$ satisfying
\begin{equation}
|f(y(0))| \le n \text { and } y'(t)=Vy(t)\text { for all } t \ge 0,
 \label{e3.5}
\end{equation}
the inequality $\|y(t)-\bar x\| \le \epsilon_1/4$ holds for some
$t \in [0,T_0]$.

\end{itemize}
Choose a positive number $\Delta$ such that
\begin{equation}
\Delta(T_0+1)e^{LT_0} \le \epsilon_1 /4 \label{e3.6}
\end{equation}
and set
\begin{equation}
\mathcal{U}=\{W \in \mathcal{A}:\; \|Wz-Vz\| \le \Delta \text { for all }
z \in B(n_1)\}.\label{e3.7}
\end{equation}
Assume that
\begin{equation}
W \in \mathcal{U}, \label{e3.8}
\end{equation}
$T \ge T_0$ and that $x \in W^{1,1}(0,T;X)$ satisfies both \eqref{e1.7}
and \eqref{e1.8}.

We show that
\begin{equation}
\|x(t)-\bar x \|\le \epsilon \text { for all } t \in [T_0,T]. \label{e3.9}
\end{equation}
By Proposition \ref{prop1.1}, there exists a differentiable mapping
$y:[0,\infty) \to X$ which satisfies
\begin{equation}
y'(t)=Vy(t),\; t \in [0,\infty), \label{e3.10}
\end{equation}
and
\begin{equation}
y(0)=x(0). \label{e3.11}
\end{equation}
In view of Proposition \ref{prop2.1} and the inclusions $V,W \in \mathcal{A}$,
the function $f \circ x$ is decreasing on $[0,T]$ and the function
$f \circ y$ is decreasing on $[0,\infty)$. Together with \eqref{e1.8} and
\eqref{e3.11}, this implies that
\begin{equation}
f(x(t)) \le n,\; t \in [0,T], \text { and }
f(y(t)) \le n,\; t \in [0,\infty). \label{e3.12}
\end{equation}
When combined with \eqref{e3.1}, this inequality implies
\begin{equation}
\|x(t)\|,\|y(t)\| \le n_1 \text { for all } t \in [0,T]. \label{e3.13}
\end{equation}
By property (P2), \eqref{e1.8}, \eqref{e3.10} and \eqref{e3.11}, there is 
a number $\tau$ such
that
\begin{equation}
\tau \in [0,T_0] \text { and } \|y(\tau)-\bar x\| \le \epsilon_1/4.
\label{e3.14}
\end{equation}
Now we will estimate $\|y(\tau)-x(\tau)\|$. It follows from 
\eqref{e1.7}, \eqref{e3.10}
and \eqref{e3.11} that for each $s \in [0,\tau]$,
\begin{equation}
\begin{aligned}
\|y(s)-x(s)\|&=\|y(0)+\int_0^sVy(t)dt-(x(0)+\int_0^sWx(t)dt)\| \\
&=\|\int_0^s(Vy(t)-Wx(t))dt\| \\
&\le \int_0^s\|Vy(t)-Wx(t)\|dt \\
&\le \int_0^s\|Vy(t)-Vx(t)\|dt+\int_0^s\|Vx(t)-Wx(t)\|dt.
\end{aligned}\label{e3.15}
\end{equation}
By \eqref{e3.13}, \eqref{e3.7} and \eqref{e3.8}, for each $s \in (0,\tau]$,
we have
\begin{equation}
\int_0^s\|Vx(t)-Wx(t)\|dt \le \int_0^s\Delta dt \le \Delta s \le
\Delta \tau. \label{e3.16}
\end{equation}
By \eqref{e3.15}, \eqref{e3.16}, \eqref{e3.13} and \eqref{e3.4}, for
each $s \in [0,\tau]$,
$$
\|y(s)-x(s)\| \le \Delta \tau+
\int_0^s\|Vy(t)-Vx(t)\|dt \le \Delta \tau+ L\int_0^s\|y(t)-x(t)\|dt.
$$
Applying Gronwall's inequality, we obtain that
$$
\|y(\tau)-x(\tau)\| \le \Delta \tau e^{\int_0^{\tau}Ldt} =\Delta \tau
e^{L\tau}.
$$
Since $\tau \le T_0$, we conclude, by using \eqref{e3.6},
that
$$
\|y(\tau)-x(\tau)\|\le \Delta T_0 e^{LT_0}<\epsilon_1/4.
$$
Together with \eqref{e3.14}, this inequality implies that
$\|x(\tau)-\bar x\| \le \epsilon_1/2$. It follows from this last
inequality and \eqref{e3.3} that $f(x(\tau)) \le f(\bar x)+\delta_1$.
Since the function
$f \circ x$ is decreasing, we infer that
$$
f(x(t)) \le f(\bar x)+\delta_1 \quad \text {for all } t \in [\tau,T].
$$
When combined with \eqref{e3.2}, this inequality
implies that
$$
\|x(t)-\bar x\| \le \epsilon \text { for all } t \in [\tau,T].
$$
The proof of Theorem \ref{thm1.2} is complete.

\subsection*{Acknowledgments}
 The second author was partially supported
by the Fund for the Promotion of Research at the Technion and by
the Technion VPR Fund - B. and G. Greenberg Research Fund
(Ottawa). Part of the third author's research was carried out when
he was visiting Ohio University.

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\end{document}