Stability analysis of split equality and split feasibility problems

TL;DR

This paper investigates the stability of solutions to split equality and split feasibility problems using set-valued and variational analysis, providing new conditions for solution map stability and illustrating their practical application.

Contribution

It introduces the first stability analysis of these problems via parametric generalized equations and establishes necessary and sufficient conditions for Lipschitz-likeness.

Findings

Derived conditions for solution map stability.

Demonstrated the application with practical examples.

Showed the necessity of nonzero solution assumption.

Abstract

In this paper, for the first time in the literature, we study the stability of solutions of two classes of feasibility (i.e., split equality and split feasibility) problems by set-valued and variational analysis techniques. Our idea is to equivalently reformulate the feasibility problems as parametric generalized equations to which set-valued and variational analysis techniques apply. Sufficient conditions, as well as necessary conditions, for the Lipschitz-likeness of the involved solution maps are proved by exploiting special structures of the problems and by using an advanced result of B.S. Mordukhovich [J. Global Optim. 28, 347--362 (2004)]. These conditions stand on a solid interaction among all the input data by means of their dual counterparts, which are transposes of matrices and regular/limiting normal cones to sets. Several examples are presented to illustrate how the obtained results work in practice and also show that the existence of nonzero solution assumption made in the necessity conditions cannot be lifted.