On Lipschitzian properties of multifunctions defined implicitly by "split" feasibility problems

TL;DR

This paper systematically analyzes the Lipschitzian properties of solution maps for split feasibility problems, providing conditions and bounds using variational analysis techniques.

Contribution

It offers a unified framework with sufficient conditions and quantitative estimates for various Lipschitzian properties of multifunctions in split feasibility problems.

Findings

Established conditions for Lipschitz lower semicontinuity and upper semicontinuity.

Derived bounds for the Lipschitz constants of the solution map.

Unified approach applicable to a broad class of convex feasibility problems.

Abstract

In the present paper, a systematic study is made of quantitative semicontinuity (a.k.a. Lipschitzian) properties of certain multifunctions, which are defined as a solution map associated to a family of parameterized ``split" feasibility problems. The latter are a particular class of convex feasibility problems with well recognized applications to several areas of engineering and systems biology. As a part of a perturbation analysis of variational systems, this study falls within the framework of a line of research pursued by several authors. It is performed by means of techniques of variational analysis, which lead to establish sufficient conditions for the Lipschitz lower semicontinuity, calmness, isolated calmness, Lipschitz upper semicontinuity and Aubin property of the solution map. Along with each of these properties, a quantitative estimate of the related exact bound is also provided. The key elements emerging on the way to achieving the main results are dual regularity conditions qualifying the problem behaviour, which are expressed in terms of convex analysis constructions involving problem data. The approach here proposed tries to unify the study of the aforementioned properties.