A condition for the stability of ideal efficient solutions in parametric vector optimization via set-valued inclusions

TL;DR

This paper investigates the stability of ideal efficient solutions in parametric vector optimization, providing conditions for their existence and stability under perturbations, with implications for solution and value mappings.

Contribution

It refines stability conditions for ideal efficient solutions in parametric vector optimization and establishes Lipschitz lower semicontinuity and calmness of related solution and value mappings.

Findings

Established Lipschitz lower semicontinuity of the solution mapping.

Proved calmness of the ideal value mapping.

Refined existence results for ideal efficient solutions.

Abstract

In present paper, an analysis of the stability behaviour of ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for the existence of ideal efficient solutions to locally perturbed problems and their nearness to a given reference value is provided by refining recent results on the stability theory of parameterized set-valued inclusions. More precisely, the Lipschitz lower semicontinuity property of the solution mapping is established, with an estimate of the related modulus. A notable consequence of this fact is the calmness behaviour of the ideal value mapping associated to the parametric class of vector optimization problems. Within such an analysis, a refinement of a recent existence result specific for ideal efficient solutions to unperturbed problem is also discussed.