Failure of Metric Regularity for Major Classes of Variational Systems

Abstract The paper is devoted to the study of metric regularity, which is a remarkable property of set-valued mappings playing an important role in many aspects of nonlinear analysis and its applications. We pay the main attention to metric regularity of the so-called parametric variational systems that contain, in particular, various classes of parameterized/perturbed variational and hemivariational inequalities, complementarity systems, sets of optimal solutions and corresponding Lagrange multipliers in problems of parametric optimization and equilibria, etc. On the basis of the advanced machinery of generalized differentiation, we surprisingly reveal that metric regularity fails for certain major classes of parametric variational systems, which admit conventional descriptions via subdifferentials of convex as well as prox-regular extended-real-valued functions.