Optimization and Optimal Control (eBook)

Prof. Pardalos is a distinguished Springer author and recognized throughout the world as a first rate mathematician. Prof. Rentsen has an extensive CV and list of publications, and he has organized the first and second International Conference for Optimization and Optimal Control (2002, 2007) in Mongolia.  (We might be able to possibly do a bulk sale for his book for the 3rd international conference.)  Pardalos, Tseveendorj, Enkhbat have edited a volume together before with World Scientific. "Optimization and Optimal Control" (World Scientific, 978-9812385970, $106, 2003) Prof. Chinchuluun has recently helped Pardalos with another optimization edited volume which will be available in 2008.

Preface 7
Contents 10
Sensibility Function as Convolution of System of Optimization Problems 13
1 Introduction 13
2 Optimization Problems for the Sensibility Function 19
3 Primal Extraproximal Method 23
3.1 Primal method 24
4 Dual Extraproximal Method 29
4.1 Dual Method 29
5 Conclusions 32
References 32
Post-optimal Analysis of Linear Semi-infinite Programs 34
1 Introduction 34
2 Preliminaries 42
2.1 Basic Concepts on Sets and Mappings 42
2.2 Basic Concepts and Results on LSIP 43
2.3 Perturbed LSIP Problems 45
3 Perturbing All the Data 47
3.1 Stability of the Feasible Set 47
3.2 Stability of the Optimal Set 49
3.3 Stability of the Optimal Value and Well-Posedness 49
3.4 Distance to Ill-Posedness 50
3.5 Error Bounds 50
3.6 Primal--Dual Stability 51
4 Perturbing the Cost Vector and the RHS Function 52
4.1 Stability Analysis 52
4.2 Sensitivity Analysis 53
5 Perturbing the RHS Function 54
5.1 Stability Analysis 54
5.2 Sensitivity Analysis 54
6 Perturbing the Cost Vector 55
6.1 Stability Analysis 55
6.2 Sensitivity Analysis 56
7 Conclusions 57
References 58
On Equilibrium Problems 65
1 Introduction 65
2 The Equilibrium Problem and Its Important Particular Cases 66
2.1 The Minimum Problem 67
2.2 The Kirszbraun's Problem 67
2.3 The Saddlepoint (Minimax Theorems) 68
2.3.1 Two-Player Zero-Sum Games 69
2.3.2 Duality in Optimization 70
2.4 Variational Inequalities 72
3 Some Existence Results on Equilibrium Problem 74
3.1 Results Based on Fixed Point Theorems 74
3.2 Results Based on Separation Theorems 79
4 The Equilibrium Problem and the Ekeland's Principle 81
4.1 The Ekeland's Principle for (EP) and (SEP) 82
4.2 New Existence Results for Equilibria on Compact Sets 87
4.3 Equilibria on Noncompact Sets 88
5 Conclusions 90
References 91
Scalarly Compactness, (S)+-Type Conditions, Variational Inequalities, and Complementarity Problems in Banach Spaces 94
1 Introduction 94
2 Preliminaries 95
3 (S)+-Type Conditions 96
4 Scalar Asymptotic Derivatives 99
5 Scalar Compactness 100
6 Existence Theorems for Variational Inequalities and Complementarity Problems 106
7 Comments 112
References 112
Quasi-equilibrium Inclusion Problemsof the Blum--Oettli-Type and Related Problems 114
1 Introduction 114
2 Preliminaries and definitions 116
3 Main Results 118
References 128
General Quadratic Programming and Its Applications in Response Surface Analysis 129
1 Introduction 129
2 Response Surface Methodology 131
3 Quadratic Convex Maximization Problem 133
4 Indefinite Quadratic Programming 135
5 Quadratic Convex Minimization Problem 138
5.1 Response Surface Practical Problems 139
6 Conclusion 144
References 144
Canonical Dual Solutions for Fixed Cost Quadratic Programs 146
1 Primal Problem and Motivation 146
2 Canonical Dual Problem 147
3 Global Optimality Criteria 152
4 Existence and Uniqueness Criteria 155
5 Application to Decoupled Problem 157
6 Examples 158
6.1 Two-Dimensional Decoupled Problem 158
6.2 General Nonconvex Problem 159
7 Concluding Remarks and Open Problems 159
References 161
Algorithms of Quasidifferentiable Optimization for the Separation of Point Sets 164
1 Introduction 164
2 Basic Notions 165
3 Quasidifferential Calculus 165
3.1 Necessary Optimality Conditions 167
4 Principal Algorithm of Finding the Intersection of Two Sets 167
5 A Minimization Method Due to Bagirov 169
6 Algorithm INTERSEC 172
7 Preliminary Numerical Results 173
References 173
A Hybrid Evolutionary Algorithmfor Global Optimization 175
1 Introduction 175
2 Evolutionary Algorithm 177
2.1 Basic Schemes 177
2.2 Procedures Used in Evolutionary Algorithm 178
3 Hybrid Evolutionary Algorithm 178
3.1 Modification of the Fitness Function 179
3.2 Population Update Rules 183
3.3 HEA Algorithm 185
4 Numerical Experiments 186
5 Conclusions 189
References 189
Gap Functions for Vector Equilibrium Problems via Conjugate Duality 191
1 Introduction 191
2 Mathematical Preliminaries 192
3 The Constrained Vector Optimization Problem 195
3.1 Different Dual Problems 195
3.2 Stability and Strong Duality 197
4 Gap Functions for Vector Equilibrium Problems 200
5 Conclusions 202
References 203
Polynomially Solvable Cases of Binary Quadratic Programs 204
1 Introduction 205
2 Problem (0--1QP) with All Off-Diagonal Elements of Q Being Non-positive 207
3 Problem (0--1QPh) with Fixed Rank Q 210
4 Problem (0--1QP) with Q Being a Tridiagonal Matrix 214
5 Problem (BQP) Defined by a Series-Parallel Graph 217
6 Problem (0--1QP) Defined by a Logic Circuit 223
7 SDP Representation of Lagrangian Dual and Polynomial Solvability 226
8 Conclusions 228
References 229
Generalized Solutions of Multi-valued Monotone Quasi-variational Inequalities 231
1 Introduction 231
2 Main Results 234
3 Applications 241
3.1 Quasi-hemivariational Inequalities 241
3.2 Inverse Problems 241
4 Concluding Remarks 242
References 242
Optimal Feedback Control for Stochastic Impulsive Linear Systems Subject to Poisson Processes 245
1 Introduction 245
2 Problem Statement 247
3 Deterministic Transformation 249
4 Time Scaling Transformation 253
5 Gradient Formulae 254
6 Example 256
7 Conclusion 261
References 261
Analysis of Differential Inclusions: Feedback Control Method 263
1 Introduction 263
2 Statement of the Problems 267
3 The Algorithm for Solving Problem 1 270
4 The Algorithm for Solving Problem 2 270
5 The Algorithm for Solving Problem 3 275
6 Conclusion 279
References 279
A Game Theoretic Algorithm to Solve Riccati and Hamilton--Jacobi--Bellman--Isaacs (HJBI) Equations in H Control 280
1 Introduction 280
2 Solving the LQ Problem by the Kleinman Algorithm 285
3 Solving H Riccati Equations 286
3.1 The Summarizing Theorem 287
3.2 Algorithm 288
3.3 Rate of Convergence 289
3.4 Game Theoretic Interpretation of the Algorithm 291
3.5 Numerical Examples 293
4 Solving an HJB Equation by a Sequenceof Linear PDEs 297
4.1 t1 Is Finite 298
4.2 t1 Is Infinite 299
5 Solving an HJBI Equation by a Sequenceof HJB Equations 301
5.1 Preliminaries and Definitions 301
5.2 The Summarizing Theorem 303
5.3 Algorithm 305
5.4 Rate of Convergence and Game Theoretic Interpretation 306
5.5 A Numerical Example 306
6 Conclusions 308
References 309
Online Adaptive Optimal Control Based on Reinforcement Learning 312
1 Introduction 312
2 The Continuous-Time Optimal Control Problem 316
3 The Policy Iteration Algorithm 317
4 Online Adaptive Optimal Control Solution Using Neural Network Elements in an Actor--Critic Structure 319
5 Simulation Results 322
6 Conclusions 324
References 325
Perturbation Methods in Optimal Control Problems 327
1 Introduction 327
2 The Perturbation Methods in Optimal Control Problems That Are Polynomial with Respect to State 329
2.1 The Optimal Control Problem That Is Polynomialwith Respect to State 329
2.2 Perturbation Method for Boundary-Value Improvement Problem 332
2.3 Transformation Method for Perturbed Boundary-Value Improvement Problems 339
2.4 Perturbation Method for Improvement Condition in Control Space 344
2.5 Projective Perturbation Method for Improvement Condition 355
3 Perturbation Methods in the Main Optimal Control Problem 357
3.1 The Main Optimal Control Problem 358
3.2 Perturbation Method for Maximum Principle 360
3.3 Projective Perturbation Method for Optimality Condition 367
3.4 Numerical Solution for Test Case 371
4 Conclusion 374
References 375
Stochastic Optimal Control with Applications in Financial Engineering 376
1 Introduction 376
2 Deterministic Optimal Control 379
2.1 Theory 379
2.2 Example: An LQ Optimal Control Problem 381
3 Stochastic Optimal Control 383
3.1 Stochastic Processes 383
3.2 Stochastic Differential Equations 384
3.3 Stochastic Calculus 385
3.4 Stochastic Optimal Control Theory 386
3.5 Example: An LQ Optimal Control Problem 387
4 Applications in Financial Engineering 389
4.1 Introduction 389
4.2 Utility Functions 390
4.3 Wealth Dynamics 391
4.3.1 Risk-Free Money Market Account 391
4.3.2 Risky Investments 391
4.3.3 Economic Influence Factors 391
4.3.4 Wealth Dynamics 392
4.4 CRRA Problems 393
4.5 CARA Problems 399
5 Conclusions 403
References 404
A Nonlinear Optimal Control Approach to Process Scheduling 410
1 Introduction 410
2 Crude Oil Scheduling Models 412
2.1 Modeling the Transfer Operation 413
2.2 Optimal Control Nonlinear Model 415
2.3 Solving the Problem 418
2.4 Mixed-Integer Linear Model 418
3 Results 419
4 Conclusion 421
References 421
Hadamard's Matrices, Grothendieck's Constant,and Root Two 423
1 Introduction 423
2 Quantum Background 424
2.1 Computational State Space 424
2.2 Entanglement 424
2.3 Observable State Space 426
2.4 SU(v) Representation 427
2.5 Amplitude Amplification 428
2.6 Correlation Matrices 429
3 The Multiknapsack Multiplayer Game Model 430
4 The Multiknapsack Quantum Simulation 432
5 The Multiknapsack Entanglement Issues 434
6 Hadamard Matrices and Fishburn and Reeds Formulation 435
7 Lehman Matrices and Grothendieck's Constant 438
8 Grothendieck's Constant and Root 2 Issues 440
8.1 2-(v,k,0=x"0115)2 Designs 440
8.2 Classical/Quantum Metaheuristics Issues 441
8.3 Quantum Oddities 442
9 Concluding Remarks 444
References 445
On the Pasture Territories Covering Maximal Grass 448
1 Main Concepts and the Problem Definition 448
2 On the Forms of Exploiting Pasture Territories in Simple Cases 451
3 Some Solution Properties of the Main Maximization Problem on the Plane 455
4 Conclusion 459
References 460
On Solvability of the Rate Control Problemin Wired-cum-Wireless Networks 461
1 Introduction 461
2 Related Works 463
3 The Rate Control Problem 464
4 Solvability of the Rate Control Problem 466
5 Numerical Example 469
6 Conclusion 471
References 476
Integer Programming of Biclustering Basedon Graph Models 477
1 Introduction 477
2 Biclustering Models 478
3 General Approach to Biclustering 479
3.1 Graph Partitioning 479
3.2 Bipartite Partitioning and Biclustering 481
4 Integer Programming of Partitioning 482
4.1 Ratio Cut 487
4.2 Normalized Cut and Minmax Cut 490
4.3 ICA Cut 493
4.4 Spectral and Integer Programming Biclustering 495
5 Conclusion 495
References 495
A Random Arrival Time Best-Choice Problem with Uniform Prior on the Number of Arrivals 497
1 Introduction 497
2 Proof of Theorem 1 500
3 Concluding Remark 507
References 507