Semi-Infinite Programming

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Rembert Reemtsen, Jan-J. Rückmann
Springer Science & Business Media, Apr 30, 1998 - Computers - 414 pages
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Semi-infinite programming (briefly: SIP) is an exciting part of mathematical programming. SIP problems include finitely many variables and, in contrast to finite optimization problems, infinitely many inequality constraints. Prob lems of this type naturally arise in approximation theory, optimal control, and at numerous engineering applications where the model contains at least one inequality constraint for each value of a parameter and the parameter, repre senting time, space, frequency etc., varies in a given domain. The treatment of such problems requires particular theoretical and numerical techniques. The theory in SIP as well as the number of numerical SIP methods and appli cations have expanded very fast during the last years. Therefore, the main goal of this monograph is to provide a collection of tutorial and survey type articles which represent a substantial part of the contemporary body of knowledge in SIP. We are glad that leading researchers have contributed to this volume and that their articles are covering a wide range of important topics in this subject. It is our hope that both experienced students and scientists will be well advised to consult this volume. We got the idea for this volume when we were organizing the semi-infinite pro gramming workshop which was held in Cottbus, Germany, in September 1996.
 

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Contents

A COMPREHENSIVE SURVEY OF LINEAR SEMIINFINITE OPTIMIZATION THEORY
3
2 EXISTENCE THEOREMS FOR THE LSIS
5
3 GEOMETRY OF THE FEASIBLE SET
6
4 OPTIMALITY
10
5 DUALITY THEOREMS AND DISCRETIZATION
12
6 STABILITY OF THE LSIS
14
7 STABILITY AND WELLPOSEDNESS OF THE LSIP PROBLEM
19
8 OPTIMAL SOLUTION UNICITY
23
3 LINEAR PROBLEMS
219
4 CONVEX PROBLEMS
234
5 NONLINEAR PROBLEMS
243
REFERENCES
262
CONNECTIONS BETWEEN SEMIINFINITE AND SEMIDEFINITE PROGRAMMING
277
2 DUALITY
280
3 ELLIPSOIDAL APPROXIMATION
281
4 EXPERIMENT DESIGN
285

REFERENCES
25
ON STABILITY AND DEFORMATION IN SEMIINFINITE OPTIMIZATION
29
2 STRUCTURE OF THE FEASIBLE SET
32
3 STABILITY OF THE FEASIBLE SET
40
4 STABILITY OF STATIONARY POINTS
44
5 GLOBAL STABILITY
53
6 GLOBAL DEFORMATIONS
57
REFERENCES
63
REGULARITY AND STABILITY IN NONLINEAR SEMIINFINITE OPTIMIZATION
69
2 UPPER SEMICONTINUITY OF STATIONARY POINTS
73
3 METRIC REGULARITY OF THE FEASIBLE SET MAPPING
83
4 STABILITY OF LOCAL MINIMIZERS
95
5 CONCLUDING REMARKS
98
REFERENCES
99
FIRST AND SECOND ORDER OPTIMALITY CONDITIONS AND PERTURBATION ANALYSIS OF SEMIINFINITE PROGRAMMING
103
2 DUALITY AND FIRST ORDER OPTIMALITY CONDITIONS
106
3 SECOND ORDER OPTIMALITY CONDITIONS
115
4 DIRECTIONAL DIFFERENTIABILITY OF THE OPTIMAL VALUE FUNCTION
122
5 STABILITY AND SENSITIVITY OF OPTIMAL SOLUTIONS
127
REFERENCES
130
PART II NUMERICAL METHODS
135
EXACT PENALTY FUNCTION METHODS FOR NONLINEAR SEMIINFINITE PROGRAMMING
137
2 EXACT PENALTY FUNCTIONS FOR SEMIINFINITE PROGRAMMING
143
LINE SEARCH ALGORITHMS
145
4 THE MULTILOCAL OPTIMIZATION SUBPROBLEM
148
5 FINAL COMMENTS
154
REFERENCES
155
FEASIBLE SEQUENTIAL QUADRATIC PROGRAMMING FOR FINELY DISCRETIZED PROBLEMS FROM SIP
159
2 ALGORITHM
163
3 CONVERGENCE ANALYSIS
167
4 EXTENSION TO CONSTRAINED MINIMAX
177
5 IMPLEMENTATION AND NUMERICAL RESULTS
180
6 CONCLUSIONS
186
PROOFS
189
NUMERICAL METHODS FOR SEMIINFINITE PROGRAMMING A SURVEY
195
2 FUNDAMENTALS
196
5 PROBLEMS INVOLVING POWER MOMENTS
289
6 POSITIVEREAL LEMMA
291
7 CONCLUSION
292
APPLICATIONS
295
RELIABILITY TESTING AND SEMIINFINITE LINEAR PROGRAMMING
297
2 TESTING SYSTEMS WITH INDEPENDENT COMPONENT FAILURES
301
3 SOLUTION PROCEDURE
306
4 TESTING SYSTEMS WITH DEPENDENT COMPONENT FAILURES
311
5 A SERIES SYSTEM WORKING IN A RANDOM ENVIRONMENT
318
6 CONCLUSIONS
320
REFERENCES
321
SEMIINFINITE PROGRAMMING IN ORTHOGONAL WAVELET FILTER DESIGN
323
A FUNCTIONAL ANALYSIS VIEW
324
2 DESIGN IMPLICATIONS FROM THE PROPERTY OF PERFECT RECONSTRUCTION
332
3 THE PERFECT RECONSTRUCTION SEMIINFINITE OPTIMIZATION PROBLEM
339
4 CHARACTERIZATION OF OPTIMAL FILTERS THROUGH SIP DUALITY
342
5 ON SOME SIP ALGORITHMS FOR QUADRATURE MIRROR FILTER DESIGN
346
6 NUMERICAL RESULTS
351
7 REGULARITY CONSTRAINTS
353
8 CONCLUSIONS
354
REFERENCES
355
THE DESIGN OF NONRECURSIVE DIGITAL FILTERS VIA CONVEX OPTIMIZATION
361
2 CHARACTERISTICS OF FIR FILTERS
364
3 APPLICATION FIELDS
368
4 APPROXIMATION PROBLEMS
371
5 THE OPTIMIZATION PROBLEM
374
6 NUMERICAL EXAMPLES
378
7 CONCLUSION
385
REFERENCES
386
SEMIINFINITE PROGRAMMING
389
1 OPTIMAL CONTROL PROBLEMS
390
2 STERILIZATION OF FOOD
395
3 FLUTTER CONTROL
401
REFERENCES
411
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