Convex Optimization · Problem 10.10

Interior-point methods — Boyd & Vandenberghe

Mathematics — Optimization · bounded-inverse-derivative-strongly-convex-concave-game

Problem

Bounded inverse derivative condition for strongly convex-concave game. Consider a convex-concave game with payoff function

f

(see page 541). Suppose

\nabla^2_{uu} f(u,v) \succeq mI

and

\nabla^2_{vv} f(u,v) \preceq -mI

, for all

(u,v) \in \mathbf{dom}\, f

. Show that

\|Dr(u,v)^{-1}\|_2 = \|\nabla^2 f(u,v)^{-1}\|_2 \leq 1/m.