Relative Lipschitz-like property of parametric systems via projectional coderivative

TL;DR

This paper develops upper estimates for the projectional coderivative of implicit mappings to analyze the relative Lipschitz-like property of solution mappings in parametric systems, with applications to variational inequalities and complementarity problems.

Contribution

It introduces a generalized critical face condition for sufficiency of Lipschitz-like properties and establishes equivalences and necessary conditions for specific classes of parametric systems.

Findings

Derived upper estimates of the projectional coderivative under various constraint qualifications.

Established a generalized critical face condition for affine variational inequalities.

Provided necessary and sufficient conditions for Lipschitz-like properties in linear complementarity problems.

Abstract

This paper concerns upper estimates of the projectional coderivative of implicit mappings and corresponding applications on analyzing the relative Lipschitz-like property. Under different constraint qualifications, we provide upper estimates of the projectional coderivative for solution mappings of parametric systems. For the solution mapping of affine variational inequalities, a generalized critical face condition is obtained for sufficiency of its Lipschitz-like property relative to a polyhedral set within its domain under a constraint qualification. The equivalence between the relative Lipschitz-like property and the local inner-semicontinuity for polyhedral multifunctions is also demonstrated. For the solution mapping of linear complementarity problems with a $Q_0$-matrix, we establish a sufficient and necessary condition for the Lipschitz-like property relative to its convex domain via the generalized critical face condition and its combinatorial nature.