Optimization and Related Topics

Applied Optimization

Volume 47

Series Editors:

Panos M. Pardalos University of Florida. US.A.

Donald Hearn University of Florida. US.A.

The titles published in this series are listed at the end o/this volume.

Optimization and Related Topics

Edited by

Alexander Rubinov School of Information Technology & Mathematical Sciences, University of Ballarat, Victoria, Australia

and

Barney Glover School of Mathematics and Statistics, Curtin University of Technology, WA, Australia

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A c.I.P. Catalogue record for this book is available from the Library of Congress.

Printed on acid-free paper

All Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

ISBN 978-1-4419-4844-1 ISBN 978-1-4757-6099-6 (eBook) DOI 10.1007/978-1-4757-6099-6

Contents

Preface

Part I NUMERICAL METHODS AND APPLICATIONS

1 AN APPROACH TO CONSTRUCTING GENERALIZED

PENALTY FUNCTIONS Mikhail Andramonov

1 Introduction 2 Generalized penalty functions for inequality constraints 3 Equality constraints

4 Modified Lagrange functions via increasing functions

xiii

3

3 4

8 10

References 14

2 AN EXACT METHOD FOR SOLVING THE SUBPROBLEM OF THE CUT- 15

TING ANGLE METHOD OF GLOBAL OPTIMIZATION Djangir A. Babayev

1 Introduction 15 2 Formulation of the problem 16 3 Lemma 17 4 Solving the subproblem when number of vectors and dimension of space

coincide 17

5 Solving the subproblem when the number of vectors is greater than dimension of space 18

6 A property of multiple solutions 24

7 Extending the presented approach to other related classes of problems 25 8 Conclusions 25

References 25

3 ON MODELING RISK IN MARKOV DECISION PROCESSES Steve Levitt, Adi Ben-Israel

1 Introduction

27

28

vi OPTIMIZATION AND RELATED TOPICS

2

3 4

5 6

Myopic optima in MDP's

A maintenance problem

A maintenance example

An inventory problem

Appendix A: The recourse certainty equivalent

References

4 MULTIPLICATIVE PROGRAMMING AND BEYOND

VIA C-PROGRAMMING Leonid Churilov, Moshe Sniedovich

1 Introduction

2 3 4

5 6 7

Parametric methods

C-Programming perspective

Multiplicative program as an additive structure

Fractional programming problems

Discussion

Summary and conclusions

References

5 COMPUTING OPTIMAL CONTROL ON MATLAB - THE SCOM PACKAGE

AND ECONOMIC GROWTH MODELS B.D. Craven, S.M.N. Islam

1 Introduction and formulation

2 Mathematical requirements for the implementation

3 Using MATLAB

4

5 Test problems

Discussion

References

6 STOCHASTIC OPTIMAL CONTROL OF A SOLAR CAR John Boland, Vladimir- Gaitsgory, Phil Howlett and Peter Pudncy

1 Introduction

2 Formulation

3 A recursive equation for the optimal controls

4 The properties of the optimal controls

5 Some elementary examples

6 Conclusions

References

7 ON OPTIMAL ALGORITHMS IN EMERGENT COMPUTATION Victor- Korotkich

1 Introduction

2 Systems of integer relations and a new type of hierarchical formations

30

31 33

34 37

39

41

42 44 47 48 53 55 57

57

61

62

63 64 64 66

67

71

71 72

73 76 77 81

81

83

83 85

Contents vii

3 4

5 6

The web of relations and structural complexity A principle specifying natural systems in the web of relations A model of computation to approximate the principle The principle in the model and coherent solutions

7 Approximation to coherent solutions and the problem of binary sequence

87 88 88 90

prediction 94 8 Constructing optimal algorithms as experimental verifications of the principle 96 9 A parameter extension of the optimal algorithm 97 10 The parameterised algorithm in combinatorial optimization 99 11 Results of computational experiments

References

8 OPTIMAL ESTIMATION OF SIGNAL PARAMETERS USING

BILINEAR OBSERVATIONS Panos M. Pardalos, Pavel S. Knopov, Stanislav P. Uryasev, Vitaliy A. Yatsenko

1 Introduction 2 3

4

5 6 7

Invertibility of continuous MS, and estimation of signal parameters Estimation of parameters of an almost periodic signal under discrete measurements Neural-network estimation of signal parameters Finite dimensional bilinear adaptive estimation Example Concluding remarks

101

101

103

104 105

109 112 114 115 116

References 116

9 ON AN EXTREMAL PROBLEM ARISING IN QUEUEING THEORY AND 119

TELECOMMUNICATIONS Michael Peake, Charles E. M. Pearce

1 Introduction 120 2 Preliminaries 121 3 Inequalities involving the auxiliary sequences 123 4

5 6

Convexity Comparison theorems Time congestion

References

10 LEVEL FUNCTIONS OF SOME OPTIMAL VALUE FUNCTIONS Huifu Xu

1

2 3 4

5 6 7

Introduction Level fu nctions Optimal value function-class 1 Optimal value function-class 2 Optimal value function-class 3 The general abstract convex setting Descent direction

125 128 131

134

135

135 137 140 141 144 146 149

viii OPTIMIZATION AND RELATED TOPICS

References 151

11 REGULARIZED GAP FUNCTIONS AND D-GAP FUNCTIONS FOR NONS- 153

MOOTH VARIATIONAL INEQUALITIES Huiju Xu

1 Introduction

2 Basics

3 Stationary points

4 Descent derivative-free method

5 Generalized Newton's method 6 Numerical tests

154

155

156

161

166 168

References 175

Part II THEORY OF OPTIMIZATION AND RELATED TOPICS

12 CONVEX SPECTRAL FUNCTIONS OF COMPACT OPERATORS, PART II: 179

LOWER SEMICONTINUITY AND REARRANGEMENT INVARIANCE Jonathan M. Borwein, Adrian S. Lewis, Qiji J. Zhu

1 Introduction 180

2 Nonexpansivity of the Eigenvalue Map 182

3 Von Neumann-type Inequalities 186

4

5 Lower Semicontinuity and Rearrangements

Lower Semicontinuity and Unitary Invariance 187 191

References 195

13 SOME INEQUALITIES FOR RIEMANN-STIELTJES 197

INTEGRAL AND APPLICATIONS S. S. Dragomir

1 Introduction 197

2 Some trapezoid like inequalities for Riemann-Stieltjes integral 199

3 Some inequalities of Ostrowski type for the Riemann-Stieltjes integral 210

4 Some inequalities of Gruss type for Riemann-Stieltjes integral 227

References 231

14 PROX-REGULARITY AND SUBJETS 237 Andrew Eberhard

1 Introduction 237

2 The inf-convolution smoothing: the first order case 242

3 A variational result for rank one representers 255

4

5 6 7

Second-order directional derivatives

The rank-one representer of a prox-regular function

Inf-convolution smoothing: the second order case

Optimality conditions for local minima

263

285

290

303

Contents ix

References 310

15 CONCERNING DIFFERENTIABILITY PROPERTIES OF LOCALLY LIPSCHITZ 315

FUNCTIONS J.R. Giles, Scott SeiJJer

1 Preliminaries 316 2 The Clarke subdifferential generated from derivatives 316 3 Upper Dini subdifferentiability 319

References 322

16 LAURENT SERIES FOR THE INVERSION OF PERTURBED LINEAR OP- 325

ERATORS ON HILBERT SPACE Phil Howlett and Kostya Avrachenkov

1 Introduction 325 2 3 4

The inverse of a perturbed matrix Inversion of perturbed linear operators on Hilbert space Inversion of perturbed linear operators on Banach space

328 330 335

References 342

17 THE EXTREMAL PRINCIPLE AND ITS APPLICATIONS TO OPTIMIZA- 343

TION AND ECONOMICS Boris S. Mordukhovich

1 Introduction 343 2 3 4

5 6

Constructions in nonsmooth analysis Extremal principle Applications to nonconvex calculus Applications to constrained optimization Applications to welfare economics

345 348 353 357 360

References 365

18 GENERIC CONVERGENCE OF INFINITE PRODUCTS OF NON EXPANSIVE 371

MAPPINGS IN BANACH AND HYPERBOLIC SPACES Simeon Reich, Alexander J. Zaslavski

1 Introduction 372 2 Hyperbolic spaces 373 3 Asymptotic behavior 4 Nonexpansive retractions 5 Convergence of Krasnosel'skii-Mann iterations 6 Contractive mappings 7 Attractive sets 8 Quasi-nonexpansive mappings 9 (F)-attracting mappings 10 Products and convex combinations of (F)-attracting

mappings and a generic result

11 Convergence of infinite products of (F)-attracting mappings

373 376 377 379 381 383 384

385

386

x OPTIMIZATION AND RELATED TOPICS

12 Extensions

13 An example of an (F)-attracting mapping

14 Convergence of random infinite products of (F)-attracting mappings

15 Regular sequences of nonexpansive mappings and their infinite Krasnosel'skii-Mann products

16 Proofs of Theorems 15.3 and 15.4

17 Auxiliary lemmas

18 Proofs of Theorems 15.5 and 15.6

References

19 RECESSION CONES OF STAR-SHAPED AND CO-STAR-SHAPED SETS A.P. Shveidel

1

2

3

Introduction

Recession cones of radiant and co-radiant sets

Recession cone of sum of two sets

References

20 DOES CONTINUITY OF CONVEX-VALUED MAPS SURVIVE

UNDER INTERSECTION? Alrxander Vladimirov

Introduction

2 3 4

5 6

7

Preliminaries

The Demyanov difference and metric

D-continuity and discontinuity of main operations on convex sets

D-regular sets

Variable D-regular sets

Applications to parametric optimization

387 388

390

390

393

395

399

400

403

403

404

412

414

415

415

416

417

420

421

424

427

References 428

21 EXISTENCE AND STRUCTURE OF SOLUTIONS OF OPTIMAL CONTROL 429

PROBLEMS Al€Xander J. Zaslavski

1 Introduction 429

2 Existence and structure of extremals of variational problems with vector-valued functions 431

3 A class of optimal control problems with time delay 434

4 Existence of overtaking optimal solutions in the class of bounded trajectories

5 Existence and asymptotic behavior of optimal solutions

6 Uniform boundedness of optimal solutions

7 The turnpike property for optimal solutions on finite intervals

8 Examples

9 A weak version of the tunpike property

10 An auxiliary result

437 438

439

439

441

448 449

11 Proof of Theorem 9.1

12 Proof of Theorem 9.2

13 Proof of Theorem 9.3

References

Contents xi

451 452 454

456

Preface

This volume contains, in part, a selection of papers presented at the sixth Australian Optimization Day Miniconference (Ballarat, 16 July 1999), and the Special Sessions on Nonlinear Dynamics and Optimization and Operations Re­search - Methods and Applications, which were held in Melbourne, July 11-15 1999 as a part of the Joint Meeting of the American Mathematical Society and Australian Mathematical Society. The editors have strived to present both con­tributed papers and survey style papers as a more interesting mix for readers. Some participants from the meetings mentioned above have responded to this approach by preparing survey and 'semi-survey' papers, based on presented lectures. Contributed paper, which contain new and interesting results, are also included.

The fields of the presented papers are very large as demonstrated by the following selection of key words from selected papers in this volume:

• optimal control, stochastic optimal control, MATLAB, economic models, implicit constraints, Bellman principle, Markov process, decision-making under uncertainty, risk aversion, dynamic programming, optimal value function.

• emergent computation, complexity, traveling salesman problem, signal estimation, neural networks, time congestion, teletraffic.

• gap functions, nonsmooth variational inequalities, derivative-free algo­rithm, Newton's method.

• auxiliary function, generalized penalty function, modified Lagrange func­tion.

• convexity, quasiconvexity, abstract convexity.

• Asplund spaces, Dini derivatives, coderivatives, subdifferential, varia­tional analysis, extremal principle, nonsmooth optimization, convex spec­tral functions, rearrangement invariant functions, composite program­ming, second order nonsmooth analysis, optimality conditions.

xiv OPTIMIZATION AND RELATED TOPICS

• global optimization, multiplicative programming, c-programming, para­metric problem, cutting angle method.

• singular perturbations, infinite dimensional linear programming, feasibil­ity problem, generic property, turnpike property, infinite horizon, over­taking optimal function.

• continuous set-valued mapping, convex-valued mappings, Hausdorff met­ric, Demyanov metric.

• radiant set, Minkowski gauge, Minkowski co-gauge, star-shaped set, re­cession cone.

• Riemann-Stieltjes integral, trapezoid inequality, midpoint inequality, Os­trowski inequality, Gruss inequality.

All contributions to this volume were carefully refereed. The editors are very grateful to the following referees: A. Bagirov (University of Ballarat, Aus­tralia), B. Craven (University of Melbourne, Australia), V. Demyanov ( State University of S.-Petersburg, Russia), A. Eberhard ( Royal Melbourne Insti­tute of Technology, Australia), Yu. Evtushenko (Computing Centre of Russian Academy of Science, Russia), J. Filar (University of South Australia, Aus­tralia), V. Gaitsgory (University of South Australia, Australia), J. Giles (Uni­versity of Newcastle, Australia), A. Ioffe (Technion, Israel), A. Jofre (Univer­sity of Chile, Chile), L. Jennings (University of Western Australia, Australia), T. Kuczumow (Marie Curie-Sklodowska University, Poland), A. Leizarowitz (Technion, Israel), W. Moors (The University of Waikato, New Zealand), C.E. M. Pearce ( The University of Adelaide, Australia), J.-P. Penot ( University of Pau, France), D. Ralph (Cambridge University, UK), M. Roughan (The Uni­versity of Melbourne, Australia), J. Sun (National University, Singapore) , J. Sunde (Defence Science and Technology Organization, Australia), J. Vander­werff (University of California, Riverside, USA), G. Wood (Massey University, New Zealand), X.Q.Yang ( Polytechnic University, Hong Kong), A. Zaffaroni (Bocconi University, Italy), A. Zaslavski (Technion, Israel).

Our special thanks to Dr. Adil Bagirov, who prepared the camera-ready copy of the manuscript.

The idea to publish this volume was supported and promoted by Prof. P. Pardalos (the editor of the series Applied Optimization) and Dr. J. Martindale (Kluwer Academic Publishers), to whom the editors and the authors are greatly indebted.

The authors also express thanks to the School of Information Technology and Mathematical Sciences, University of Ballarat, Victoria, Australia, for provid­ing facilities and support for preparing the camera-ready copy of this volume.

A.M. RUBINOV AND B.M. GLOVER