Progress in Optimization: Contributions from Australasia

347 Pages·2000·12.508 MB·English

by  Hoang Tuy (auth.), Dr. Xiaoqi Yang Ph. D., Alistair I. Mees, Mike Fisher, Les Jennings (eds.)

Progress in Optimization
Applied Optimization
Volume 39
Series Editors:
Panos M. Pardalos
University of Florida, U.SA.
Donald Hearn
University of Florida, U.S.A.
The titles published in this series are listed at the end of this volume.
Progress in Optimization
Contributions from Australasia
edited by
Xiaoqi Yang
The Hong Kong Polytechnic University,
Hong Kong
Alistair I. Mees
Mike Fisher
and
Les Jennings
The University o/Western Australia,
Perth, Australia
....
"
KLUWER ACADEMIC PUBLISHERS
DORDRECHT/BOSTON/LONDON
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-13: 978-1-4613-7986-7 e-ISBN-13: 978-1-4613-0301-5
DOl: 10.1007/978-1-4613-0301-5
Published by Kluwer Academic Publishers,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
Sold and distributed in North, Central and South America
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved
© 2000 Kluwer Academic Publishers
Softcover reprint of the hardcover 1st edition 2000
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,
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retrieval system, without written permission from the copyright owner
Contents
Preface xi
Participants xv
Editors xix
Part I GLOBAL OPTIMIZATION
1
GLOBAL OPTIMIZATION METHODS FOR LOCATION AND DISTANCE 3
GEOMETRY PROBLEMS
Hoang Thy
1 Introduction 4
2 Global Optimization 5
3 A Generic BB Method 6
4 The Generalized Weber's Problem 8
5 Various Objectives in Facility Location 9
6 Constrained Location 12
7 Challenging Problems 14
8 Conclusions 17
References 17
2
BRANCH AND CUT METHODS FOR MIXED INTEGER LINEAR PROGRAM- 21
MING PROBLEMS
Louis Caccetta
1 Introduction 22
2 Solution Methods 23
3 The Traveling Salesman Problem 28
4 Vehicle Routing Problem 30
5 A Mine Scheduling Problem 35
6 Conclusions 39
References 39
3
SEPARABILITY OF STAR-SHAPED SETS WITH RESPECT TO INFINITY 45
A. M. Rubinov and A. P. Shveidel
1 Introduction 46
2 Preliminaries 47
3 Strongly Star-Shaped Sets 48
v
vi PROGRESS IN OPTIMIZATION
4 Separation 52
5 Sum of Strongly Star-Shaped Sets w.r.t. Infinity 60
6 Separation of Strongly Star-Shaped Sets w.r.t. Infinity 62
4
NONLINEAR UNCONSTRAINED OPTIMIZATION METHODS: A REVIEW 65
A. M. Rubinov, X. Q. Yang and B. M. Glover
1 Introduction 66
2 Nonlinear Penalty Functions 67
3 Penalty Functions for Problems with One Constraint 69
4 Nonlinear Lagrangian Functions 70
5 Lagrange Functions for Problems with One Constraint 72
6 Penalty and Lagrange 73
7 Parameter Nonlinear Unconstrained Optimization Methods 74
References 77
5
NEW DUAL FORMULATIONS IN CONSTRAINED INTEGER PROGRAMMING 79
Xiaoling Sun and Duan Li
1 Introduction 80
2 Duality Gap in the Classical Lagrangian Method 80
3 p-norm Surrogate Constraint Method 84
4 p-th Power Lagrangian Method 85
5 Minimax-type Lagrangian Function 86
6 Logarithmic-exponential Dual Formulation 87
7 Conclusions 90
References 90
6
SIMULATED ANNEALING AND PENALTY METHODS FOR BINARY MUL- 93
TICOMMODITY FLOW PROBLEMS
X.Q. Yang, A.I. Mees and K. Campbell
1 Introduction 94
2 The Binary M ulticommodity Flow Problem 95
3 Penalty Functions for Discrete Problems 96
4 Simulated Annealing for Discrete Multicommodity Flows 100
5 Results 103
6 Conclusions 104
References 104
Part II NONSMOOTH OPTIMIZATION
7
A QUADRATIC RECOURSE FUNCTION FOR THE TWO-STAGE STOCHAS- 109
TIC PROGRAM
John R. Birge, Stephen M. Pollock and Liqun Qi
Contents vii
1 In trod uction 110
2 The Recourse Function 111
3 Differentiability of the Quadratic Recourse Function 113
4 Approximation to the Linear Recourse Function 115
5 Conclusions 120
References 120
8
LAGRANGE MULTIPLIERS FOR NONCONVEX OPTIMIZATION 123
B. D. Craven
1 Introduction 124
2 Necessary Conditions 125
3 Sufficient Conditions 127
4 Duality 127
References 128
9
CLASS-INCLUSION PROPERTIES FOR CONVEX FUNCTIONS 129
Andreu' Eberhard and Charles E. M. Pearce
1 Introduction 130
2 Proper Class Inclusions 131
3 Lower Semicontinuity 132
References 132
10
ON GENERIC LOCALLY CONVEX VECTOR FUNCTIONS 135
V. Gershkovich, B.D. Craven and D. Ralph,
1 In trod uction 136
2 Regular and Critical Points of Vector Functions 137
3 Generic LC Vector Functions F : Mn ~ IRk with 2k - 4 < n 141
4 On Locally Convex Vector Functions with Small Singularities 147
5 Future Research 150
References 151
11
ESSENTIAL COMPONENTS AND CONNECTEDNESS OF SOLUTION SET 153
FOR COMPLEMENTARITY PROBLEMS
George [sac and George X. Z. Yuan
1 Introduction 154
2 Preliminaries 154
3 Complementarity Problems 155
4 Connectedness of Solution Set 155
5 Essential Connected Components of Solution Set for Complementarity Prob-
lems 157
6 Applications to Complementarity Theory 159
7 The General Case 161
viii PROGRESS IN OPTIMIZATION
12
ON RELATIONS BETWEEN VECTOR VARIATIONAL INEQUALITY AND 167
VECTOR OPTIMIZATION PROBLEM
Gue Myung Lee
1 Introduction 168
2 Minty Type Vector Variational Inequality 168
3 Weak Vector Variational Inequality 173
4 Existence Theorems for Vector Optimization Problem 176
References 178
Part III OPTIMIZATION METHODS
13
PARAMETER ESTIMATION IN DYNAMIC SYSTEMS 183
Klaus SchittkoU'ski
1 Introduction 184
2 Data Fitting Methods 185
3 Systems of Ordinary Differential Equations 189
4 Systems of Differential Algebraic Equations 191
5 Systems of One-Dimensional Time-Dependent Partial Differential Equations 193
6 Systems of One-Dimensional Partial Differential Algebraic Equations 197
7 Applications 199
8 Conclusions 200
References 200
14
METHODS OF FEASIBLE DIRECTIONS: A REVIEW 205
Xibin Chen and Michael M. Kostreva
1 Introduction 206
2 Typical Methods of Feasible Directions 207
3 The Comparisons of Three MFD 214
4 Concluding Remarks 217
References 218
15
COMPUTATIONAL METHOD FOR A CLASS OF OPTIMAL SWITCHING 221
CONTROL PROBLEMS
Y. Liu and K.L. Teo
1 Introduction 222
2 Problem Formulation 223
3 Problem Transformation 224
4 Gradient Formulae 233
5 Example 233
6 Conclusions 236
References 237
Contents ix
16
OPTIMIZATION BY WAY OF THE TRAJECTORY FOLLOWING METHOD 239
Thomas L. Vincent
1 Introduction 240
2 Trajectory Fo"owing Algorithm for Simple Constraints 242
3 Non-Simple Constraints 244
4 Linear/Quadratic Programming 246
5 Global Minimization 249
6 Conclusions 254
References 254
17
SOLVING HAMILT ON-JACOB I-BELLMAN EQUATIONS BY AN UPWIND 255
FINITE DIFFERENCE METHOD
S. Wang, F. Gao and K.L. Teo
1 Introduction 257
2 The Method 258
3 Stability of the Scheme 260
4 Numerical Experiments 264
5 Conclusions 268
References 268
18
AN EFFICIENT APPROXIMATION METHOD FOR A CLASS OF CONTIN- 269
UOUS LINEAR PROGRAMS
K. H. Wong, M. I. Kasiama and C. Myburgh
1 Introduction 270
2 Statement of the Problem 270
3 Approximate Problem 271
4 Analysis of the Method 274
5 Piecewise Linear Continuous Approximation 276
6 Constraint Approximation 278
7 Convergence Result 280
8 Example 282
References 284
Part IV APPLICATIONS
19
CALIBRATION OF PARAMETERS FOR A COMBINED GRAVITY AND TRAF- 287
FIC ASSIGNMENT MODEL
Renlong Han
1 Introduction 288
2 Proposed Model 289
3 Algorithms 292
4 Model Tests 293
5 Results 296