Optimization Methods and Applications

By: Xiao-qi Yang (Editor), Kok Lay Teo (Editor), Lou Caccetta (Editor)

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Hardcover | 30 April 2001

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This edited book is dedicated to Professor N. U. Ahmed, a leading scholar and a renowned researcher in optimal control and optimization on the occasion of his retirement from the Department of Electrical Engineering at University of Ottawa in 1999. The contributions of this volume are in the areas of optimal control, non linear optimization and optimization applications. They are mainly the im proved and expanded versions of the papers selected from those presented in two special sessions of two international conferences. The first special session is Optimization Methods, which was organized by K. L. Teo and X. Q. Yang for the International Conference on Optimization and Variational Inequality, the City University of Hong Kong, Hong Kong, 1998. The other one is Optimal Control, which was organized byK. ~Teo and L. Caccetta for the Dynamic Control Congress, Ottawa, 1999. This volume is divided into three parts: Optimal Control; Optimization Methods; and Applications. The Optimal Control part is concerned with com putational methods, modeling and nonlinear systems. Three computational methods for solving optimal control problems are presented: (i) a regularization method for computing ill-conditioned optimal control problems, (ii) penalty function methods that appropriately handle final state equality constraints, and (iii) a multilevel optimization approach for the numerical solution of opti mal control problems. In the fourth paper, the worst-case optimal regulation involving linear time varying systems is formulated as a minimax optimal con trol problem.

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Preface	p. xi
An Appreciation of Professor N.U. Ahmed	p. xiii
A Publication List of Professor N.U. Ahmed	p. xvii
Optimal Control
Practical Stability of Impulsive Delay Differential Equations and Applications to Control Problems	p. 3
Introduction	p. 4
Preliminaries	p. 4
Main Results	p. 6
Application	p. 10
References	p. 21
A Review of Ill-Conditioning and Regularization in Optimal Control Computation	p. 23
Introduction	p. 24
Optimal Control Problem Template for Miser3	p. 25
Control Parametrization	p. 26
State Discretization Methods	p. 27
Condition Numbers for Constrained Optimization	p. 30
Test Problem 1	p. 32
Regularization of Optimal Control Problems	p. 35
Test Problem 2: The Container Crane Problem	p. 39
Conclusions	p. 40
References	p. 42
Worst-Case Optimal Regulation of Linear Systems in the Presence of Structured Perturbations	p. 45
Introduction	p. 46
Problem Statement	p. 47
Optimal Solution	p. 50
Computation of Disturbance Rejection Capacity	p. 57
Examples	p. 58
Conclusions	p. 61
References	p. 61
True Proportional Navigation System with Acceleration Saturation Constraint	p. 65
Introduction	p. 66
System Model	p. 66
Saturation Constraint Analysis	p. 68
Observability with Saturation Constraint	p. 74
Simulation Results	p. 76
Conclusions	p. 79
References	p. 79
Evaluation of Penalty Functions for Optimal Control	p. 81
Introduction	p. 82
Problem Formulation	p. 82
Iterative Dynamic Programming (IDP)	p. 84
Numerical Results	p. 86
Concluding Remarks	p. 100
References	p. 102
On the Optimal Control Systems with Multiple Controllers	p. 105
Introduction	p. 106
Optimal Regulator and Preliminaries	p. 107
LQR Problem with 2 Controllers	p. 109
Controllers Decoupling via Internally Balanced State Space Representations	p. 115
Numerical Examples and Discussions	p. 118
Conclusion	p. 122
References	p. 123
Multilevel Optimization of Optimal Control Problems	p. 125
Introduction	p. 126
General Problem	p. 128
Multilevel Approach	p. 128
Implementation	p. 131
Numerical Results	p. 133
Parallel Implementation	p. 136
Conclusions	p. 137
References	p. 138
Modified Direct Gradient Descent Control of Nonlinear Systems	p. 139
Introduction	p. 140
Direct Gradient Descent Control	p. 140
Modified Direct Gradient Descent Control	p. 142
Simulation	p. 144
Concluding Remarks	p. 146
References	p. 146
Stability	p. 148
Computation of Feedback Control for Infinite Time Optimal Control Problems	p. 151
Introduction	p. 152
Problem Statement	p. 153
The Approximation Problem (P(T))	p. 153
Synthesizing the Optimal Feedback Control Law for the Approximate Problem (P(T))	p. 154
Interpolation using a Cubic Spline Approach on the Fitting Domain D[subscript T]	p. 156
Creating an Approximate Problem Using Cubic Spline Feedback Structure on the Domain D[subscript T]	p. 157
Properties of Asymptotic Stability of the Optimal Control Feedback Control of the Problem (Q)	p. 158
Finite Time Approximation to the Problem (Q)	p. 159
A Practical Example	p. 159
References	p. 163
Optimization Methods
A Homogenized Cutting Plane Method to Solve the Convex Feasibility Problem	p. 167
Introduction	p. 168
Barrier functions and proximity measures	p. 169
A column generation method	p. 171
Choosing a restart point	p. 175
Global convergence	p. 178
An implementation	p. 179
Computational results	p. 185
Conclusions	p. 188
References	p. 189
Algorithms for Some Hard Knapsack Problems	p. 191
Introduction	p. 192
Preliminaries	p. 195
Subset Sum Problem (SSP)	p. 197
Strongly Correlated Problem (SCKP)	p. 201
Inverse Strongly Correlated Knapsack Problem (ISCKP)	p. 205
Bounded Knapsack Problem (BKP)	p. 207
Conclusions	p. 214
References	p. 214
Non-Static Network Optimization Problems: A Survey	p. 219
Introduction	p. 220
The Non-Static Network Flow Model	p. 220
Non-Static Shortest Path Problems	p. 224
Non-Static Maximum Flow Problems	p. 227
Non-Static Minimum Cost Flow Problems	p. 230
Non-Static Vehicle Routing Problems	p. 233
Other Non-Static Network Optimization Problems	p. 236
Conclusions	p. 237
References	p. 238
Asymptotic Rates of Convergence of Sqp-Type Methods of Feasible Directions	p. 247
Introduction	p. 248
Definitions and Propositions	p. 250
Asymptotic Rate of Convergence	p. 254
Comparison of Convergence Rates	p. 262
Conclusions	p. 263
References	p. 264
Nonlinear Lagrangian Methods in Constrained Nonlinear Optimization	p. 267
Introduction	p. 268
p-th Power Lagrangian Method	p. 270
Minimax-type Lagrangian Function	p. 271
Logarithmic-Exponential Lagrangian Function	p. 274
Conclusions	p. 276
References	p. 276
Parallel Algorithms for Solving Large-Scale Nonlinear Optimization Problems	p. 279
Conventional Methods for Nonlinear Optimization	p. 280
Parallel Quasi-Newton Algorithms	p. 281
Vectorization and Fine-tuning Techniques	p. 284
Computational Results	p. 288
References	p. 293
Second Order Strict Converse Duality in Nonlinear Fractional Programming	p. 295
Introduction and Preliminaries	p. 296
Second Order Strict Converse Duality I	p. 298
Second Order Strict Converse Duality II	p. 302
References	p. 305
Optimization Applications
Chebyshev Optimization of Circular Arrays	p. 309
Introduction	p. 310
Problem Formulation	p. 311
The Complex Chebyshev Approximation Problem as a Semi-Infinite Linear Program	p. 312
Semi-Infinite Linear Programming	p. 313
Brief Outline of Linear Programming	p. 315
Numerical Example	p. 316
Summary	p. 317
References	p. 317
Optimum Pole Position for Digital Laguerre Network with Least Square Error Criterion	p. 321
Introduction	p. 322
Problem Formulation	p. 322
Optimization Method	p. 324
Simulation Studies	p. 326
Conclusions	p. 328
References	p. 329
A Markov Model for the Stochastic Optimal Control of a Solar Powered Car	p. 331
Introduction	p. 332
Formulation	p. 332
A Recursive Equation for the Optimal Controls	p. 333
The Properties of the Optimal Controls	p. 336
Some Elementary Examples	p. 338
Conclusions	p. 341
References	p. 341
The Personnel Task Scheduling Problem	p. 343
Introduction	p. 344
The PTSP	p. 344
Variants of the PTSP	p. 348
Literature Review	p. 357
Applications	p. 359
Test Data	p. 362
Conclusions	p. 366
References	p. 367
Envelope Constrained Filter Design: Robustness Issues	p. 369
Introduction	p. 370
Envelope Constrained Filtering	p. 372
Robustness Issues	p. 380
Conclusions	p. 395
References	p. 395
A Robust Numerical Algorithm for the Optimal Control of Heat Transfer in the Continuous Casting of Steel	p. 399
Introduction	p. 400
Heat Transfer Model	p. 401
Statement of the Optimal Control Problem	p. 403
Numerical Method for the Optimal Control Problem	p. 404
Numerical Approximation of the Jacobian Matrix	p. 406
Numerical Algorithm	p. 407
An Illustrative Example	p. 408
Conclusions	p. 410
References	p. 411
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Optimization Methods and Applications

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