Rosen / Mangasarian / Ritter Nonlinear Programming

Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin, Madison, May 4-6, 1970
1. Auflage 2014
ISBN: 978-1-4832-7246-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 502 Seiten, Web PDF

Nonlinear Programming contains the proceedings of a Symposium on Nonlinear Programming held in Madison, Wisconsin on May 4-6, 1970. This book emphasizes algorithms and related theories that lead to efficient computational methods for solving nonlinear programming problems. This compilation consists of 17 chapters. Chapters 1 to 9 are concerned primarily with computational algorithms, while Chapters 10 to 13 are devoted to theoretical aspects of nonlinear programming. Certain applications of nonlinear programming are considered in Chapters 14 to 17. The algorithms for nonlinear constraint problems, investigation of convergence rates, and use of nonlinear programming for approximation are also covered in this text. This publication is a good source for students and researchers concerned with nonlinear programming.

Autoren/Hrsg.

Rosen, J. B.

Mangasarian, O. L.

Ritter, K.

Inhaltsverzeichnis

1;Front Cover;1
2;Nonlinear Programming;4
3;Copyright Page;5
4;Table of Contents;6
5;Foreword;10
6;Preface;12
7;Chapter 1. A Method of Centers by Upper-Bounding Functions with Applications;14
7.1;ABSTRACT;14
7.2;Introduction;15
7.3;1. The Method of Centers: A Summary with Modifications;17
7.4;2. Method of Centers (General algorithm;21
7.5;3. Method of Center by Upper-Bounding Functions;25
7.6;4. Applications of the Method of Centers by Upper- Bounding
Functions;31
7.7;REFERENCES;42
8;Chapter 2. A New Algorithm for Unconstrained Optimization;44
8.1;ABSTRACT;44
8.2;1. Introduction;45
8.3;2. The Formula for Revising the Second Derivative
Approximation;47
8.4;3. An Outline of the New Algorithm;50
8.5;4. Theorems on the New Algorithm;55
8.6;Acknowledgements;75
8.7;REFERENCES;77
9;Chapter 3. A Class of Methods for Nonlinear Programming
II Computational Experience;80
9.1;ABSTRACT;80
9.2;Introduction;81
9.3;2. A Basic Approach;87
9.4;3. Algorithms based on Variable Metric methods;91
9.5;4. Inequality Constraints;102
9.6;REFERENCES;104
10;Chapter 4. Some Algorithms Based on the
Principle of Feasible Directions;106
10.1;ABSTRACT;106
10.2;1. Introduction;107
10.3;2. Direction generators;107
10.4;3. Unconstrained Optimization;117
10.5;4. Linearly Constrained Nonlinear Programming;122
10.6;5. A partitioning method;128
10.7;REFERENCES;133
11;Chapter 5. Numerical Techniques in Mathematical Programming;136
11.1;ABSTRACT;136
11.2;Introduction;138
11.3;A. THE USE OF LU DECOMPOSITION IN
EXCHANGE ALGORITHMS;139
11.4;B. THE QR DECOMPOSITION AND
QUADRATIC PROGRAMMING;155
11.5;C. THE SVD AND NONLINEAR LEAST
SQUARES;173
11.6;REFERENCES;185
12;Chapter 6. A Superlinearly Convergent Method for
Unconstrained Minimization;190
12.1;ABSTRACT;190
12.2;1. Introduction;191
12.3;2. Formulation of the problem, definitions and notation;191
12.4;3. The algorithm;193
12.5;4. Special convergence properties of the algorithm;203
12.6;REFERENCES;219
13;Chapter 7. A Second Order Method for the Linearly Constrained
Nonlinear Programming Problem;220
13.1;ABSTRACT;220
13.2;1. Introduction;221
13.3;2. The algorithm;226
13.4;3. Convergence of the Algorithm;233
13.5;4. Rate of Convergence of the Algorithm;248
13.6;5. Discussion;254
13.7;REFERENCES;254
14;Chapter 8. Convergent Step-Sizes for Gradient-Like Feasible
Direction Algorithms for Constrained Optimization;258
14.1;ABSTRACT;258
14.2;1. Introduction;259
14.3;2. Gradient-like feasible direction algorithms;260
14.4;3. General stepsize criteria;262
14.5;4. Step sizes based on minimization;266
14.6;5. Step sizes based on a range function;271
14.7;6. Step sizes based on a search procedure;276
14.8;7 • Example of directions: variable metric gradient
projections;281
14.9;REFERENCES;285
15;Chapter 9. On the Implementation of Conceptual Algorithms;288
15.1;ABSTRACT;288
15.2;1. Introduction;289
15.3;2. Conceptual algorithms;290
15.4;3. Adaptive Procedures for Implementation;296
15.5;4. Open Loop Procedures for Implementation;300
15.6;5. Conclusion;304
15.7;REFERENCES;304
16;Chapter 10. Some Convex Programs Whose Duals
Are Linearly Constrained;306
16.1;ABSTRACT;306
16.2;1. Introduction;307
16.3;2. Dual problems;308
16.4;3. The nature of problem
(D1);312
16.5;4. Examples;318
16.6;5. Relationships between (P), (D )
and (DI);326
16.7;REFERENCES;333
17;Chapter 11. Sufficiency Conditions and a Duality Theory
for Mathematical Programming Problems in Arbitrary Linear Spaces;336
17.1;ABSTRACT;336
17.2;1. Introduction;337
17.3;2. Mathematical preliminaries and problem statement;338
17.4;3. Necessary conditions and sufficient conditions;341
17.5;4. Duality;349
17.6;5. An application to the theory of optimal control;354
17.7;REFERENCES;360
18;Chapter 12. Recent Results on Complementarity Problems;362
18.1;ABSTRACT;362
18.2;Introduction;363
18.3;I. The Linear Complementarity Problem;364
18.4;II. Nonlinear Complementarity Problems;384
18.5;APPENDIX;392
18.6;REFERENCES;395
19;Chapter 13. Nonlinear Nondifferentiable
Programming in Complex Space;398
19.1;ABSTRACT;398
19.2;Preliminaries;399
19.3;Introduction;399
19.4;REFERENCES;412
20;Chapter 14. Duality Inequalities of Mathematics and Science;414
20.1;ABSTRACT;414
20.2;1. Introduction;415
20.3;2. Finite linear programming;416
20.4;3. A duality gap;417
20.5;4. Infinite programming;418
20.6;5. Application of infinite programming to analysis;420
20.7;6. Infinite programming and convex programmingConsider the convex constraint;422
20.8;7. The extremal length and width of a network;423
20.9;8. Optimum heat transfer;425
20.10;9. Upper and lower networks;426
20.11;10. Problems which are not self-adjoint;428
20.12;11. Legendre transforms and nonlinear networks;430
20.13;12. Chemical equilibrium;432
20.14;13. Schrodinger's equation;433
20.15;REFERENCES;434
21;Chapter 15. Programming Methods in Statistics
and Probability Theory;438
21.1;ABSTRACT;438
21.2;1. Introduction;439
21.3;2. The structure of most powerful tests;440
21.4;3. A new optimality concept;443
21.5;4. An application of geometric programming;444
21.6;5. Tchebycheff bounds;447
21.7;6. Unbiased estimators with smallest variance;454
21.8;REFERENCES;456
22;Chapter 16. Applications of Mathematical Programming to
lP Approximation;460
22.1;ABSTRACT;460
22.2;I. Introduction;461
22.3;II. Best
lp approximation;462
22.4;III. Linear, convex, and concave programming;463
22.5;IV. Mathematical programming and
lp approximation;464
22.6;V. Algorithms;468
22.7;VI. Test examples;474
22.8;VII. Conclusions;475
22.9;Acknowledgements;475
22.10;REFERENCES;477
23;Chapter 17. Theoretical and Computational Aspects
of Nonlinear Regression;478
23.1;ABSTRACT;478
23.2;1. Introduction;479
23.3;2. Iterative Methods for Nonlinear Regression;481
23.4;3. Theoretical Comparison;485
23.5;4. Computational Devices;494
23.6;5. Computational Experience;495
23.7;Acknowledgement;498
23.8;REFERENCES;498
24;Index;500