Convex Optimization · Problem 4.54

Duality — Boyd & Vandenberghe

Mathematics — Optimization · scalarization-optimal-points

Problem

Scalarization and optimal points. Suppose a (not necessarily convex) vector optimization problem has an optimal point x⋆x^\star. Show that x⋆x^\star is a solution of the associated scalarized problem for any choice of λ≻K∗0\lambda \succ_{K^*} 0. Also show the converse: If a point xx is a solution of the scalarized problem for any choice of λ≻K∗0\lambda \succ_{K^*} 0, then it is an optimal point for the (not necessarily convex) vector optimization problem.

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