Metric regularity is among the central concepts of nonlinear and variational
analysis, constrained optimization, and their numerous applications. However, met-
ric regularity can be elusive for some important ill-posed classes of problems includ-
ing 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 anal-
ysis, 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 differenti-
ation 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.
Article