Bridging local and semilocal stability: A topological approach

TL;DR

This paper introduces a topological condition linking local and semilocal stability of set-valued mappings, enabling precise calculation of error bounds in various non-convex optimization frameworks.

Contribution

It establishes a general topological condition that relates local and semilocal stability, extending stability analysis to non-convex mappings and enabling exact semilocal error bound calculations.

Findings

Lipschitz upper semicontinuity modulus equals the supremum of local calmness moduli under certain conditions

The theorem applies to non-convex frameworks like semi-algebraic mappings and generalized equations

Enables precise, point-based calculation of semilocal error bounds

Abstract

This paper establishes a general topological condition under which the semilocal stability of a set-valued mapping can be exactly determined by its local stability properties. Specifically, we investigate the relationship between the Lipschitz upper semicontinuity modulus -- a semilocal measure of variation for the image set -- and the local calmness moduli. While these two quantities coincide for mappings with convex graphs, the relationship generally breaks down in the absence of convexity, making the semilocal modulus exceptionally difficult to compute. We prove that if a mapping is outer semicontinuous in the Painlev\'{e}-Kuratowski sense and locally compact around the nominal parameter, the Lipschitz upper semicontinuity modulus is exactly the supremum of the local calmness moduli over the nominal set. In addition to the theoretical advance, this equality enables the precise calculation of semilocal error bounds via point-based formulae. We illustrate the broad applicability of this theorem by setting it up in several non-convex frameworks in parametric optimization, including piecewise convex and semi-algebraic mappings, feasible and optimal set mappings under full data perturbations, generalized equations and linear complementarity problems, semi-infinite inequality systems, and parameterized sub-level sets.