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Bridging local and semilocal stability: A topological approach

J. Camacho

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 Painlevé-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.

Bridging local and semilocal stability: A topological approach

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 Painlevé-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.
Paper Structure (9 sections, 9 theorems, 44 equations)

This paper contains 9 sections, 9 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Classes of mappings and applications
  5. Piecewise Convex and Semi-Algebraic Systems
  6. Parameterized Generalized Equations and LCPs
  7. Linear Semi-Infinite Inequality Systems
  8. Parameterized Sub-Level Set Mappings
  9. Conclusions

Key Result

Theorem 1

Let $\mathcal{M}:Y\rightrightarrows X$ be a set-valued mapping between metric spaces. Let $\overline{y}\in\mathop{\mathrm{dom}}\nolimits\mathcal{M}$ such that $\mathcal{M}$ is outer semicontinuous in the Painlevé-Kuratowski sense at $\overline{y}$. Furthermore, assume that $\mathcal{M}$ is locally c

Theorems & Definitions (19)

  • Remark 1
  • Theorem 1
  • Corollary 1
  • Remark 2
  • Example 1: Lack of Closedness
  • Example 2: Lack of Outer Semicontinuity
  • Example 3: Lack of Local boundedness
  • Proposition 1
  • Theorem 2
  • Corollary 2
  • ...and 9 more