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Preface | ||
1 | Optimization Problem Formulation | 1 |
1.1 | Optimization Problem Formulation | 2 |
1.2 | The Standard Form of an Optimization Problem | 13 |
1.3 | Solution of Optimization Problems | 16 |
1.4 | Time Value of Money | 18 |
2 | Graphical Optimization | 47 |
2.1 | Procedure for Graphical Optimization | 48 |
2.2 | Graphical Solution Function | 57 |
2.3 | Graphical Optimization Examples | 59 |
3 | Mathematical Preliminaries | 75 |
3.1 | Vectors and Matrices | 75 |
3.2 | Approximation Using the Taylor Series | 89 |
3.3 | Solution of Nonlinear Equations | 100 |
3.4 | Quadratic Forms | 106 |
3.5 | Convex Functions and Convex Optimization Problems | 112 |
4 | Optimality Conditions | 131 |
4.1 | Optimality Conditions for Unconstrained Problems | 132 |
4.2 | The Additive Property of Constraints | 144 |
4.3 | Karush-Kuhn-Tucker (KT) Conditions | 147 |
4.4 | Geometric Interpretation of KT Conditions | 165 |
4.5 | Sensitivity Analysis | 175 |
4.6 | Optimality Conditions for Convex Problems | 181 |
4.7 | Second-Order Sufficient Conditions | 187 |
4.8 | Lagrangian Duality | 199 |
5 | Unconstrained Problems | 227 |
5.1 | Descent Direction | 229 |
5.2 | Line Search Techniques - Step Length Calculations | 231 |
5.3 | Unconstrained Minimization Techniques | 253 |
6 | Linear Programming | 315 |
6.1 | The Standard LP Problem | 316 |
6.2 | Solving a Linear System of Equations | 319 |
6.3 | Basic Solutions of an LP Problem | 334 |
6.4 | The Simplex Method | 339 |
6.5 | Unusual Situations Arising During the Simplex Solution | 365 |
6.6 | Post-Optimality Analysis | 376 |
6.7 | The Revised Simplex Method | 387 |
6.8 | Sensitivity Analysis Using the Revised Simplex Method | 402 |
7 | Interior Point Methods | 437 |
7.1 | Optimality Conditions for Standard LP | 438 |
7.2 | The Primal Affine Scaling Method | 445 |
7.3 | The Primal-Dual Interior Point Method | 464 |
Appendix | Null and Range Spaces | 481 |
8 | Quadratic Programming | 495 |
8.1 | KT Conditions for Standard QP | 495 |
8.2 | The Primal Affine Scaling Method for Convex QP | 502 |
8.3 | The Primal-Dual Method for Convex QP | 520 |
8.4 | Active Set Method | 535 |
8.5 | Active Set Method for the Dual QP Problem | 552 |
Appendix | Derivation of the Descent Direction Formula for the PAS Method | 563 |
9 | Constrained Nonlinear Problems | 581 |
9.1 | Normalization | 582 |
9.2 | Penalty Methods | 585 |
9.3 | Linearization of a Nonlinear Problem | 608 |
9.4 | Sequential Linear Programming - SLP | 614 |
9.5 | Basic Sequential Quadratic Programming - SQP | 620 |
9.6 | Refined SQP Methods | 645 |
Appendix | An Introduction to Mathematica | 677 |
Bibliography | 705 | |
Index | 709 |
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