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On the role of semismoothness in nonsmooth numerical analysis: Theory

H. Gfrerer, J. V. Outrata

Abstract

For the numerical solution of nonsmooth problems, sometimes it is not necessary that an exact subgradient/generalized Jacobian is at our disposal, but it suffices that a semismooth derivative, i.e., a mapping satisfying a certain semismoothness property, is available. In this paper we consider not only semismooth derivatives of single-valued mappings, but also its interplay with the semismoothness$^*$ property for multifunctions. In particular, we are interested in semismooth derivatives of solution maps to parametric semismooth$^*$ inclusions. Our results are expressed in terms of suitable generalized derivatives of the set-valued part, i.e., by limiting coderivatives or by SC (subspace containing) derivatives. Further we show that semismooth derivatives coincide a.e. with generalized Jacobians and state some consequences concerning strict proto-differentiability for semismooth$^*$ multifunctions.

On the role of semismoothness in nonsmooth numerical analysis: Theory

Abstract

For the numerical solution of nonsmooth problems, sometimes it is not necessary that an exact subgradient/generalized Jacobian is at our disposal, but it suffices that a semismooth derivative, i.e., a mapping satisfying a certain semismoothness property, is available. In this paper we consider not only semismooth derivatives of single-valued mappings, but also its interplay with the semismoothness property for multifunctions. In particular, we are interested in semismooth derivatives of solution maps to parametric semismooth inclusions. Our results are expressed in terms of suitable generalized derivatives of the set-valued part, i.e., by limiting coderivatives or by SC (subspace containing) derivatives. Further we show that semismooth derivatives coincide a.e. with generalized Jacobians and state some consequences concerning strict proto-differentiability for semismooth multifunctions.
Paper Structure (11 sections, 21 theorems, 94 equations)

This paper contains 11 sections, 21 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Background from variational analysis and preliminary results
  3. Nonsmooth analysis
  4. Variational geometry and differentiation of multifunctions
  5. Closure of mappings
  6. SCD mappings
  7. On semismooth$^*$ mappings
  8. On $\Psi$-semismooth single-valued mappings
  9. On strict proto-differentiability and the SCD-ss$^*$ property of graphically Lipschitzian mappings
  10. On the semismooth$^*$ property of solution maps to inclusions
  11. Conclusion

Key Result

Lemma 2.1

Let $\Omega\subset\mathbb{R}^n$ be open and let $F:\Omega\to\mathbb{R}^m$ be Lipschitz near $\bar{x}\in \Omega$. Then the following statements are equivalent:

Theorems & Definitions (50)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7: cf. RoWe98
  • Lemma 2.8: cf. GfrOut23
  • ...and 40 more