Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2012 | Book

Introduction Read chapter Read first chapter

Topological Aspects of Nonsmooth Optimization

Author: Vladimir Shikhman

Publisher: Springer New York

Book Series : Springer Optimization and Its Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

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. ​

​

Table of Contents

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
Metadata
Title
Topological Aspects of Nonsmooth Optimization
Author
Vladimir Shikhman
Copyright Year
2012
Publisher
Springer New York
Electronic ISBN
978-1-4614-1897-9
Print ISBN
978-1-4614-1896-2
DOI
https://doi.org/10.1007/978-1-4614-1897-9