Convex Optimization · Problem 4.53

Duality — Boyd & Vandenberghe

Mathematics — Optimization · convex-vector-optimization-achievable-set

Problem

Suppose the vector optimization problem (4.56) is convex. Show that the set

\mathcal{A} = \mathcal{O} + K = \{t \in \mathbf{R}^q \mid f_0(x) \preceq_K t \text{ for some feasible } x\},

is convex. Also show that the minimal elements of

\mathcal{A}

are the same as the minimal points of

\mathcal{O}