Marginal Analysis of Convex Optimization Problems with Set-Valued Inclusion Constraints

TL;DR

This paper analyzes the stability and sensitivity of convex optimization problems with set-valued inclusion constraints, providing Lipschitz continuity results and subgradient formulas under convexity assumptions.

Contribution

It introduces a novel sensitivity analysis framework for parametric convex problems with set-valued constraints using variational analysis techniques.

Findings

Optimal value function is Lipschitz continuous under certain conditions.

Subgradient formulas for the optimal value function are derived.

Stability properties inform penalization and calmness analysis.

Abstract

In this paper, stability and sensitivity properties of a class of parametric constrained optimization problem, whose feasible region is defined by a set-valued inclusion, are investigated through the associated optimal value function. Set-valued inclusions are a kind of constraint system, which naturally emerges in contexts requiring the robust fulfilment of traditional cone constraints, where data are affected by uncertain elements having a non stochastic nature, or in (MPEC) as a vector equilibrium constraint, where feasible solutions are intended as equilibrium point in a strong sense. Under proper convexity assumptions on the objective function and the constraining set-valued term, combined with a global qualification condition, a class of parametric optimization problems is singled out, which displays a global Lipschitz behaviour. By employing recent results of variational analysis, elements for a sensitivity analysis of this class of problems are provided via exact subgradient formulae for the optimal value function. Further consequences of the stability behaviour are explored in terms of problem calmness and viability of penalization techniques.