Computational Methods in Optimization (eBook)

Front Cover 1
Computational Methods in Optimization: A Unified Approach 4
Copyright Page 5
Contents 8
Preface 10
Note to the Reader 14
Conventions and Symbols 16
CHAPTER 1. PRELIMINARY RESULTS 20
1.1 Nonlinear Programming and Optimal Control Problems 20
1.2 Optimality Conditions 26
1.3 Models and Convergence Conditions for Computational Methods 31
CHAPTER 2. UNCONSTRAINED MINIMIZATION 47
2.1 Gradient and Quasi-Newton Methods in Rn 47
2.2 Reduction of Derivative Calculations 59
2.3 Conjugate Gradient Methods in Rn 63
2.4 Unconstrained Discrete Optimal Control Problems 85
2.5 Unconstrained Continuous Optimal Control Problems 90
CHAPTER 3. EQUALITY CONSTRAINTS: PROBLEMS ROOT AND BOUNDARY-VALUE 98
3.1 Zeros of a Function and Problems with Equality Constraints in Rn 98
3.2 Boundary-Value Problems and Discrete Optimal Control 102
3.3 Boundary-Value Problems and Continuous Optimal Control 122
CHAPTER 4. EQUALITY AND INEQUALITY CONSTRAINTS 145
4.1 Penalty Function Methods 145
4.2 Methods of Centers 169
4.3 Methods of Feasible Directions 178
4.4 Second-Order Methods of Feasible Directions 199
4.5 Gradient Projection Methods 204
CHAPTER 5. CONVEX CONTROL OPTIMAL PROBLEMS 227
5.1 Nonlinear Programming Algorithms Revisited 227
5.2 A Decomposition Algorithm of the Dual Type 230
5.3 A Decomposition Algorithm of the Primal Type 253
CHAPTER 6. RATE OF CONVERGENCE 261
6.1 Linear Convergence 261
6.2 Superlinear Convergence: Quasi-Newton Methods 270
6.3 Superlinear Convergence: Conjugate Gradient Methods 278
6.4 Superlinear Convergence: the Variable Metric Algorithm 287
APPENDIX A: FURTHER MODELS FOR COMPUTATIONAL METHODS 302
A.1 A Model for the Implementation of Certain Conceptual Optimal Control Algorithms 302
A.2 An Open-Loop Model for the Implementation of Conceptual Algorithms 307
APPENDIX B: PROPERTIES O F CONTINUOUS FUNCTIONS 311
B.1 Expansions of Continuous Functions 311
B.2 Convex Functions 313
8.3. A Few Miscellaneous Results 314
APPENDIX C: A GUIDE TO IMPLEMENTABLE ALGORITHMS 318
C.1 General Considerations 318
C.2 Gradient Methods 320
C.3 Quasi-Newton Methods 323
C.4 Conjugate Gradient Algorithms 325
C.5 Penalty Function Methods 328
C.6 Methods of Feasible Directions with Linear Search 331
C.7 Methods of Feasible Directions with Quadratic Search 334
References 336
Index 342
Mathematics in Sciecne and Engineering 350