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This book deals with nonsmooth structures arising within the optimization setting. It considers four optimization problems, namely, mathematical programs with complementarity constraints, general semi-infinite programming problems, mathematical programs with vanishing constraints and bilevel optimization. The author uses the topological approach and topological invariants of corresponding feasible sets are investigated. Moreover, the critical point theory in the sense of Morse is presented and parametric and stability issues are considered. The material progresses systematically and establishes a comprehensive theory for a rather broad class of optimization problems tailored to their particular type of nonsmoothness.

Topological Aspects of Nonsmooth Optimization will benefit researchers and graduate students in applied mathematics, especially those working in optimization theory, nonsmooth analysis, algebraic topology and singularity theory. ​

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
We state mathematical programming problems with complementarity constraints, general semi-infinite programming problems, mathematical programming problems with vanishing constraints and bilevel optimization. The topological approach for studying problems above is introduced. It encompasses the study of topological properties of corresponding feasible sets, as well as the critical point theory in the sense of Morse. Finally, we describe the application of the topological approach for standard nonlinear programming problems.
Vladimir Shikhman

Chapter 2. Mathematical Programming Problems with Complementarity Constraints

Abstract
We study mathematical programming problems with complementarity constraints (MPCC) from the topological point of view. The (topological) stability of the MPCC feasible set is addressed. Therefore, we introduce Mangasarian-Fromovitz condition (MFC) and its stronger version (SMFC). Under SMFC, the MPCC feasible set is shown to be a Lipschitz manifold. The links to other well-known constraint qualifications for MPCCs are elaborated. The critical point theory for MPCCs is presented. We also characterize the strong stability of C-stationary points for MPCC, dealing with parametric aspects for MPCCs.
Vladimir Shikhman

Chapter 3. General Semi-infinite Programming Problems

Abstract
We study general semi-infinite programming problems (GSIP) from the topological point of view. Introducing the symmetric Mangasarian-Fromovitz constraint qualification (Sym-MFCQ) for GSIPs, we describe the closure of the GSIP feasible set. It is proved that Sym-MFCQ is stable and generic. Moreover, under Sym-MFCQ, the GSIP feasible set is shown to be a Lipschitz manifold. For GSIPs, we state the nonsmooth symmetric reduction ansatz (NSRA). NSRA is proven to hold generically at all KKT points for the GSIP. NSRA allows us to reduce the GSIP to a so-called disjunctive optimization problem. This reduction enables to establish the critical point theory for GSIPs.
Vladimir Shikhman

Chapter 4. Mathematical Programming Problems with Vanishing Constraints

Abstract
We study mathematical programming problems with vanishing constraints (MPVC) from the topological point of view. The critical point theory for MPVCs is presented. For that, we introduce the notion of a T-stationary point for the MPVC.
Vladimir Shikhman

Chapter 5. Bilevel Optimization

Abstract
We study bilevel optimization problems from the optimistic perspective. The case of one-dimensional leader’s variable is considered. Based on the classification of generalized critical points in one-parametric optimization, we describe the generic structure of the bilevel feasible set. Moreover, optimality conditions for bilevel problems are stated. In the case of higher-dimensional leader’s variable, some bifurcation phenomena are discussed. The links to the singularity theory are elaborated.
Vladimir Shikhman

Chapter 6. Impacts on Nonsmooth Analysis

Abstract
We discuss the notions of regular and critical points/values for nonsmooth functions. The notion of topologically regular points for min-type functions is introduced. It is shown that the level set of a min-type function corresponding to a regular value, is a Lipschitz manifold. The application of the Clarke’s implicit function theorem is discussed here. A nonsmooth version of Sard’s Theorem for min-type functions is shown. Finally, we discuss the links to metrically regular and critical points/values of nonsmooth functions.
Vladimir Shikhman

Backmatter

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