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No-gap second-order conditions for minimization problems in spaces of measures

Gerd Wachsmuth, Daniel Walter

Abstract

Over the last years, minimization problems over spaces of measures have received increased interest due to their relevance in the context of inverse problems, optimal control and machine learning. A fundamental role in their numerical analysis is played by the assumption that the optimal dual state admits finitely many global extrema and satisfies a second-order sufficient optimality condition in each one of them. In this work, we show the full equivalence of these structural assumptions to a no-gap second-order condition involving the second subderivative of the Radon norm as well as to a local quadratic growth property of the objective functional with respect to the bounded Lipschitz norm.

No-gap second-order conditions for minimization problems in spaces of measures

Abstract

Over the last years, minimization problems over spaces of measures have received increased interest due to their relevance in the context of inverse problems, optimal control and machine learning. A fundamental role in their numerical analysis is played by the assumption that the optimal dual state admits finitely many global extrema and satisfies a second-order sufficient optimality condition in each one of them. In this work, we show the full equivalence of these structural assumptions to a no-gap second-order condition involving the second subderivative of the Radon norm as well as to a local quadratic growth property of the objective functional with respect to the bounded Lipschitz norm.
Paper Structure (19 sections, 35 theorems, 233 equations)

This paper contains 19 sections, 35 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related work
  4. Outline
  5. Notation
  6. Minimization problems in spaces of measures
  7. No-gap second-order conditions for abstract minimization problems
  8. No-gap second-order conditions in spaces of measures
  9. Uniform local Quasiconvexity
  10. A primer on the dual space of $C^1(\Omega)$ and the relation to the bounded Lipschitz norm
  11. A lifted setting
  12. The main result
  13. Structural assumptions imply SSC
  14. Twice strong-strong epi-differentiability of $H$
  15. Twice weak* epi-differentiability of $G$
  16. ...and 4 more sections

Key Result

Lemma 3.2

Let ass:functions hold. Then the operator $K \colon \mathcal{M}(\Omega) \to Y$ as defined in def:integraloperator is linear and sequentially weak*-to-strong continuous. Moreover, we have $K=(K_*)^*$ where the preadjoint $K_* \colon Y \to C(\Omega)$ is given by

Theorems & Definitions (73)

  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Definition 4.2
  • Theorem 4.3
  • Definition 4.4
  • Lemma 4.5
  • Lemma 4.6
  • proof
  • Definition 5.1
  • ...and 63 more