Convex Optimization · Problem 2.16

Convex functions — Boyd & Vandenberghe

Mathematics — Optimization · partial-sum-convex-sets

Problem

Show that if

S_1

and

S_2

are convex sets in

\mathbf{R}^{m+n}

, then so is their partial sum

S = \{(x, y_1 + y_2) \mid x \in \mathbf{R}^m, \; y_1, y_2 \in \mathbf{R}^n, \; (x, y_1) \in S_1, \; (x, y_2) \in S_2\}.