The Least Singular Value Function in Variational Analysis

Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, metric regularity can be elusive for some important ill-posed classes of problems including polynomial equations, parametric variational systems, smooth reformulations of complementarity systems with degenerate solutions, etc. The study of stability issues for such problems can often not rely on the machinery of first-order variational analysis, and so higher-order regularity concepts have been proposed in recent years. In this paper, we investigate some notions of mixed-order regularity by using advanced tools of first-order and second-order variational analysis and generalized differentiation of both primal and dual types. Efficient characterizations of such mixed-order regularity concepts are established by employing a fresh notion of the least singular value function. The obtained conditions are applied to deriving constructive criteria for mixed-order regularity in coupled constraint and variational systems.