Some results on condition numbers in convex multiobjective optimization

TL;DR

This paper introduces new condition numbers for convex multiobjective optimization problems, compares local and global approaches, and extends classical distance theorems to this context.

Contribution

It proposes novel condition numbers for multiobjective problems and compares local and global perspectives, extending classical theorems to this setting.

Findings

Both condition numbers reduce to Zolezzi's scalar case.

A pseudodistance bounds function perturbations for well-conditioning.

Extended Eckart-Young theorem for specific perturbations.

Abstract

Various notions of condition numbers are used to study some sensitivity aspects of scalar optimization problems. The aim of this paper is to introduce a notion of condition number to study the case of a multiobjective optimization problem defined via m convex C^1,1 objective functions on a given closed ball in R^n. Two approaches are proposed: the first one adopts a local point of view around a given solution point, whereas the second one considers the solution set as a whole. A comparison between the two notions of well-conditioned problem is developed. We underline that both the condition numbers introduced in the present work reduce to the same of condition number proposed by Zolezzi in 2003, in the special case of the scalar optimization problem considered there. A pseudodistance between functions is defined such that the condition number provides an upper bound on how far from a well-conditioned function f a perturbed function g can be chosen in order that g is well-conditioned too. For both the local and the global approach an extension of classical Eckart-Young distance theorem is proved, even if only a special class of perturbations is considered.