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\input{../../_assets/basicslide1}
\input{../../_assets/macros}

\author{Samuel Vaiter}
\date{}
\institute{
Created: 2023-11-08.\\
Last update: 2023-11-08.\\
Status: inprogress.}
\title{Local minima of convex functions are global}

\begin{document}

\begin{frame}
  \titlepage
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\begin{frame}
  \frametitle{Local minima of convex functions are global}
  \begin{equation}\label{eq:minpb}\tag{Opt}
    \min_{x \in C} f(x) ,
  \end{equation}
  \begin{theorem}
    Assume that $f: C \to \RR$ is a convex function and $C$ is a convex \textbf{closed} set.
    Then,
    \begin{enumerate}
    \item\label{local-global} Any local solution of \eqref{eq:minpb} is a global solution;
    \item\label{global-convex} The set of global solutions of \eqref{eq:minpb} is convex.
    \end{enumerate}
  \end{theorem}
  \metroset{block=transparent}
  \begin{block}{Proof idea}
    \begin{enumerate}
    \item By contradiction.
    \item Consider two solutions and study the segment joining them.
    \end{enumerate}
  \end{block}
  \metroset{block=fill}
\end{frame}

%% TODO: full proof
    
\end{document}