Optimization Theory and Methods: Nonlinear Programming

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Springer Science & Business Media, Aug 6, 2006 - Mathematics - 688 pages
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Optimization Theory and Methods can be used as a textbook for an optimization course for graduates and senior undergraduates. It is the result of the author's teaching and research over the past decade. It describes optimization theory and several powerful methods. For most methods, the book discusses an idea’s motivation, studies the derivation, establishes the global and local convergence, describes algorithmic steps, and discusses the numerical performance.

 

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Contents

612 Convergence of TrustRegion Methods
308
613 Solving A TrustRegion Subproblem
316
62 Conic Model and Collinear Scaling Algorithm
324
622 Generalized QuasiNewton Equation
326
623 Updates that Preserve Past Information
330
624 Collinear Scaling BFGS Algorithm
334
63 Tensor Methods
337
632 Tensor Methods for Unconstrained Optimization
341

131 Convex Sets
32
132 Convex Functions
36
133 Separation and Support of Convex Sets
50
14 Optimality Conditions for Unconstrained Optimization
57
15 Structure of Optimization Methods
63
Exercises
68
Line Search
71
22 Convergence Theory for Exact Line Search
74
23 The Golden Section Method and the Fibonacci Method
84
232 The Fibonacci Method
87
24 Interpolation Method
89
242 Cubic Interpolation Method
98
25 Inexact Line Search Techniques
102
251 Armijo and Goldstein Rule
103
252 WolfePowell Rule
104
253 Goldstein Algorithm and WolfePowell Algorithm
106
254 Backtracking Line Search
108
255 Convergence Theorems of Inexact Line Search
109
Exercises
116
Newtons Methods
119
312 Convergence of the Steepest Descent Method
120
313 Barzilai and Borwein Gradient Method
126
Kantorovich Inequality
129
32 Newtons Method
130
33 Modified Newtons Method
135
34 FiniteDifference Newtons Method
140
35 Negative Curvature Direction Method
147
351 GillMurray Stable Newtons Method
148
352 FiaccoMcCormick Method
151
353 FletcherFreeman Method
152
354 SecondOrder Step Rules
155
36 Inexact Newtons Method
163
Exercises
172
Conjugate Gradient Method
174
42 Conjugate Gradient Method
178
422 Beales ThreeTerm Conjugate Gradient Method
185
423 Preconditioned Conjugate Gradient Method
188
43 Convergence of Conjugate Gradient Methods
191
432 Convergence Rate of Conjugate Gradient Methods
198
Exercises
200
QuasiNewton Methods
203
511 QuasiNewton Equation
204
512 Symmetric RankOne SR1 Update
207
513 DFP Update
210
514 BFGS Update and PSB Update
217
515 The Least Change Secant Update
223
52 The Broyden Class
225
53 Global Convergence of QuasiNewton Methods
231
531 Global Convergence under Exact Line Search
232
532 Global Convergence under Inexact Line Search
238
54 Local Convergence of QuasiNewton Methods
240
541 Superlinear Convergence of General QuasiNewton Methods
241
542 Linear Convergence of General QuasiNewton Methods
250
543 Local Convergence of Broydens RankOne Update
255
544 Local and Linear Convergence of DFP Method
258
545 Superlinear Convergence of BFGS Method
261
546 Superlinear Convergence of DFP Method
265
547 Local Convergence of Broydens Class Methods
271
55 SelfScaling Variable Metric SSVM Methods
273
552 SelfScaling Variable Metric SSVM Method
277
553 Choices of the Scaling Factor
279
56 Sparse QuasiNewton Methods
282
57 Limited Memory BFGS Method
292
Exercises
301
TrustRegion Methods and Conic Model Methods
302
Exercises
349
Solving Nonlinear LeastSquares Problems
353
72 GaussNewton Method
355
73 LevenbergMarquardt Method
362
732 Convergence of LevenbergMarquardt Method
367
74 Implementation of LM Method
372
75 QuasiNewton Method
379
Exercises
382
Theory of Constrained Optimization
384
82 FirstOrder Optimality Conditions
388
83 SecondOrder Optimality Conditions
401
84 Duality
406
Exercises
409
Quadratic Programming
411
92 Duality for Quadratic Programming
413
93 EqualityConstrained Quadratic Programming
419
94 Active Set Methods
427
95 Dual Method
435
96 Interior Ellipsoid Method
441
97 PrimalDual InteriorPoint Methods
445
Exercises
451
Penalty Function Methods
455
102 The Simple Penalty Function Method
461
103 Interior Point Penalty Functions
466
104 Augmented Lagrangian Method
474
105 Smooth Exact Penalty Functions
480
106 Nonsmooth Exact Penalty Functions
482
Exercises
490
Feasible Direction Methods
493
112 Generalized Elimination
502
113 Generalized Reduced Gradient Method
509
114 Projected Gradient Method
512
115 Linearly Constrained Problems
515
Exercises
520
Sequential Quadratic Programming
522
122 WilsonHanPowell Method
530
123 Superlinear Convergence of SQP Step
537
124 Maratos Effect
541
125 Watchdog Technique
543
126 SecondOrder Correction Step
545
127 Smooth Exact Penalty Functions
550
128 Reduced Hessian Matrix Method
554
Exercises
558
TrustRegion Methods for Constrained Problems
561
132 Linear Constraints
563
133 TrustRegion Subproblems
568
134 Null Space Method
571
135 CDT Subproblem
580
136 PowellYuan Algorithm
585
Exercises
594
Nonsmooth Optimization
597
142 Nonsmooth Optimization Problem
607
143 The Subgradient Method
609
144 Cutting Plane Method
615
145 The Bundle Methods
617
146 Basic Property of a Composite Nonsmooth Function
620
147 Trust Region Method for Composite Nonsmooth Optimization
623
148 Nonsmooth Newtons Method
628
Exercises
634
Test Functions
636
2 Test Functions for Constrained Optimization Problems
638
Bibliography
649
Index
682
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