Nonconvex Optimization and Its Applications

By: Oliver Stein

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Hardcover | 31 August 2003

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236 Pages

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Semi-infinite optimization is a vivid field of active research. Recently semi infinite optimization in a general form has attracted a lot of attention, not only because of its surprising structural aspects, but also due to the large number of applications which can be formulated as general semi-infinite programs. The aim of this book is to highlight structural aspects of general semi-infinite programming, to formulate optimality conditions which take this structure into account, and to give a conceptually new solution method. In fact, under certain assumptions general semi-infinite programs can be solved efficiently when their bi-Ievel structure is exploited appropriately. After a brief introduction with some historical background in Chapter 1 we be gin our presentation by a motivation for the appearance of standard and general semi-infinite optimization problems in applications. Chapter 2 lists a number of problems from engineering and economics which give rise to semi-infinite models, including (reverse) Chebyshev approximation, minimax problems, ro bust optimization, design centering, defect minimization problems for operator equations, and disjunctive programming.

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"This is the first book which exploits the bilevel structure of semi-infinite programming systematically. It highlights topological and structural aspects of general semi-infinite programming, formulates powerful optimality conditions, and gives a conceptually new bilevel solution method. The book is addressed to graduate students and researchers who work in the fields of optimization and operations research." (Oliver Stein, Zentralblatt MATH, Vol. 1103 (5), 2007)

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Non-Fiction Mathematics Optimisation Linear Programming Computing & I.T.Computer Programming & Software Development Applied Mathematics Geometry Analytic Geometry Discrete Mathematics Calculus & Mathematical Analysis Numerical Analysis

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Published: 26th November 2013

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List of Symbols	p. xi
List of Figures	p. xv
List of Tables	p. xix
Preface	p. xxi
Acknowledgments	p. xxvii
Introduction	p. 1
Standard semi-infinite programming	p. 2
General semi-infinite programming	p. 4
The misconception about the generality of GSIP	p. 6
Development to a field of active research	p. 8
Examples and Applications	p. 11
Chebyshev and reverse Chebyshev approximation	p. 12
Minimax problems	p. 14
Robust optimization	p. 15
Design centering	p. 18
Defect minimization for operator equations	p. 20
Disjunctive programming	p. 22
Topological Structure of the Feasible Set	p. 25
Abstract index set mappings	p. 25
A projection formula	p. 27
A bi-level formula and semi-continuity properties	p. 31
A set-valued mapping formula	p. 41
The local structure of M	p. 42
The completely convex case	p. 44
Index set mappings with functional constraints	p. 46
The convex case	p. 46
The linear case	p. 47
The C[superscript 1] case	p. 60
The C[superscript 2] case	p. 62
Genericity results	p. 66
Optimality Conditions	p. 85
Abstract primal optimality conditions	p. 85
First order approximations of the feasible set	p. 90
General constraint qualifications	p. 91
Descriptions of the linearization cones	p. 96
Degenerate index sets	p. 108
Dual first order optimality conditions	p. 116
The standard semi-infinite case	p. 118
The completely convex case	p. 120
The convex case	p. 123
The C[superscript 2] case with Reduction Ansatz	p. 126
The C[superscript 1] case	p. 128
Second order optimality conditions	p. 142
Bi-Level Methods for GSIP	p. 145
Reformulations of GSIP	p. 146
The Stackelberg game reformulation of GSIP	p. 146
The MPEC reformulation of GSIP	p. 148
A regularization of MPEC by NCP functions	p. 149
The regularized Stackelberg game	p. 152
Convergence results for a bi-level method	p. 154
A parametric reduction lemma	p. 155
Convergence of global solutions	p. 157
Convergence of Fritz John points	p. 158
Quadratic convergence of the optimal values	p. 162
An outer approximation property	p. 163
Other bi-level approaches and generalizations	p. 167
Computational Results	p. 171
Design centering in two dimensions	p. 172
Design centering in higher dimensions	p. 177
Robust optimization	p. 178
Optimal error bounds for an elliptic operator equation	p. 181
Final Remarks	p. 187
References	p. 191
Index	p. 201
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Nonconvex Optimization and Its Applications

At a Glance

Hardcover

Industry Reviews

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You Can Find This Book In

Other Editions and Formats

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