Convex Optimization · Problem 2.24

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · supporting-hyperplanes

Problem

Supporting hyperplanes. ((a)) Express the closed convex set

\{x \in \mathbf{R}^2_+ \mid x_1 x_2 \geq 1\}

as an intersection of halfspaces. ((b)) Let

C = \{x \in \mathbf{R}^n \mid \|x\|_\infty \leq 1\}

, the

\ell_\infty

-norm unit ball in

\mathbf{R}^n

, and let

\hat{x}

be a point in the boundary of

C

. Identify the supporting hyperplanes of

C

\hat{x}

explicitly.