On Some Regularity Properties in Variational Analysis

Abstract

Some properties, connected with recent generalizations of the classic notion of Lipschitz continuity for multifunctions, are investigated with reference to variational systems, that is to solution maps associated to parametrized generalized equations. The latter ones are a convenient framework to address several questions, mainly related to the stability and sensitivity analysis, arising in mathematical programming, optimal control, equilibrium and variational inequality theory. Global and local criteria for metric regularity and Lipschitz-likeness of variational systems are obtained. Some applications to the exact penalization of mathematical programs with equilibrium constraints and to the Lipschitzian stability of fixed points for multivalued contractions are then considered.

Existence and continuity of solution trajectories of generalized equations with application in electronics

“…Continuity of Solution Trajectories. In this subsection we will discuss the relation between the input signal and solution trajectories under the strong metric regularity assumption of the auxiliary map (24). Throughout the whole subsection we will assume that, given a function p(·), a solution trajectory z(·) exists.…”

Stability and Sensitivity Analysis of Solutions to Weak Vector Variational Inequalities

“…The reader can refer to Example 3.1 of [57]. Since calmness implies continuity, the assumptions (iii) and (iv) can be replaced by the calmness of g at (p,x).…”

“…Recently, there are many papers to study a class of parametric variational systems, see [1,2,38,57]. Aragón Artacho and Mordukhovich [1,2] investigated the metric regularity and Lipschitz-likeness of an especial parametric variational system, where the field mapping only depends on the decision variable.…”

“…Aragón Artacho and Mordukhovich [1,2] investigated the metric regularity and Lipschitz-likeness of an especial parametric variational system, where the field mapping only depends on the decision variable. Not long ago, Uderzo [57], Li and Li [38] and Li et al [31] studied respectively a parametric variational system, where the field mapping both depends on the decision variable and the parameter variable. Uderzo [57] discussed the global and local criteria for Lipschitzian property and metric regularity of the parametric variational system.…”

“…Not long ago, Uderzo [57], Li and Li [38] and Li et al [31] studied respectively a parametric variational system, where the field mapping both depends on the decision variable and the parameter variable. Uderzo [57] discussed the global and local criteria for Lipschitzian property and metric regularity of the parametric variational system. Li and Li [38] studied the Robinson's metric regularity and the contingent derivative for the parametric variational system.…”