EDB — 188

Exercises

1. [188] Topics:subdifferential.Prerequisites:[184]↺↻.Difficulty:*.

   Let $C\subseteq {ℝ}^n$ be an open convex set, and $f:C\to ℝ$ a convex function; Given $z\in C$, we define the subdifferential $\partial f(z)$ as the set of $v$ for which the relation [(14.24)]↺↻ is valid (i.e., $\partial f(z)$ contains all vectors $v$ defining the support planes to $f$ in $z$).

   $\partial f(z)$ enjoys interesting properties.

   • $\partial f(z)$ is locally bounded: if $z\in C$ and $r>0$ is such that $B(z,2r)\subset C$, then $L>0$ exists such that $\forall y\in B(z,r)$, $\forall v\in \partial f(x)$ you have $|v|\le L$. In particular, for every $z\in C$, we have that $\partial f(z)$ is a bounded set.

   • Show that $\partial f$ is upper continuous in this sense: if $z\in C$ and $(z_n{)}_n\subset C$ and $v_n\in \partial f(z_n)$ and if $z_n{\to }_{n}z$ and $v_n{\to }_{n}v$ then $v\in \partial f(z)$. In particular, for every $z\in C$, $\partial f(z)$ is a closed set.

   Solution 1

   [189]↺↻