On metric subregularity for convex constraint systems by primal equivalent conditions

Abstract

In this paper, we mainly study metric subregularity for a convex constraint system defined by a convex set-valued mapping and a convex constraint subset. The main work is to provide several primal equivalent conditions for metric subregularity by contingent cone and graphical derivative. Further it is proved that these primal equivalent conditions can characterize strong basic constraint qualification of convex constraint system given by

Primal criteria of metric subregularity for generalized equations

“…This property is proved to closely relate with error bounds, linear regularity, and basic constraint qualification (BCQ) in optimization and have a huge range of applications in areas of variational analysis and mathematical programming like optimality conditions, variational inequalities, subdifferential theory, the sensitivity analysis of generalized equations and convergence analysis of algorithms for solving equations or inclusions. For these reasons, the concept of the metric subregularity has been extensively studied by many authors (see, e.g., [9,10,11,12,13,14,15,16,17] and the references therein).…”

“…It is noted that a pretty natural idea is to study the metric subregularity in terms of various primal derivative-like objects, such as directional derivatives, contingent cones, or slopes. Several criteria for metric subregularity of generalized equations were presented in [1,9,10,11,25,26,27,28,29,30,31] dependent on the primal-type estimate. Based on the works from [9,30,31] that are to give primal characterizations of metric subregularity for the convex generalized equation, a natural issue is to extend primal results on metric subregularity by dropping the convexity assumption.…”

“…Several criteria for metric subregularity of generalized equations were presented in [1,9,10,11,25,26,27,28,29,30,31] dependent on the primal-type estimate. Based on the works from [9,30,31] that are to give primal characterizations of metric subregularity for the convex generalized equation, a natural issue is to extend primal results on metric subregularity by dropping the convexity assumption. Inspired by this issue, our goal in the paper is to discuss primal criteria of metric subregularity for the generalized equation (not necessarily convex).…”

On Constraint qualifications of a nonconvex inequality

“…Subsequently Zheng and Ng [34] discussed BCQ and strong BCQ for a convex constraint system and applied strong BCQ to prove dual characterizations for metric subregularity of convex constraint system. Recently Huang and Wei [18] studied metric subregularity of convex constraint system by primal equivalent conditions and demonstrated that these primal conditions can characterize strong BCQ for this case (cf. [18, Theorem 3.1 and Propoition 4.1]).…”