EDB — 188

Exercises

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

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

   \(∂ f(z)\) enjoys interesting properties.

   • \(∂ f(z)\) is locally bounded: if \(z∈ C\) and \(r{\gt}0\) is such that \(B(z,2r)⊂ C\), then \(L{\gt}0\) exists such that \(∀ y∈ B(z,r)\), \(∀ v∈ ∂ f(x)\) you have \(|v|≤ L\). In particular, for every \(z∈ C\), we have that \(∂ f(z)\) is a bounded set.

   • Show that \(∂ f\) is upper continuous in this sense: if \(z∈ C\) and \((z_ n)_ n⊂ C\) and \(v_ n∈∂ f(z_ n)\) and if \(z_ n→_ n z\) and \(v_ n→_ n v\) then \(v∈∂ f(z)\). In particular, for every \(z∈ C\), \(∂ f(z)\) is a closed set.

   Solution 1

   [189]↺↻