Conjugate Gradient Algorithms in Nonconvex Optimization

By: Radoslaw Pytlak

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Hardcover | 11 December 2008

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Conjugate direction methods were proposed in the early 1950s. When high speed digital computing machines were developed, attempts were made to lay the fo- dations for the mathematical aspects of computations which could take advantage of the ef?ciency of digital computers. The National Bureau of Standards sponsored the Institute for Numerical Analysis, which was established at the University of California in Los Angeles. A seminar held there on numerical methods for linear equationswasattendedbyMagnusHestenes, EduardStiefel andCorneliusLanczos. This led to the ?rst communication between Lanczos and Hestenes (researchers of the NBS) and Stiefel (of the ETH in Zurich) on the conjugate direction algorithm. The method is attributed to Hestenes and Stiefel who published their joint paper in 1952 [101] in which they presented both the method of conjugate gradient and the conjugate direction methods including conjugate Gram-Schmidt processes. A closelyrelatedalgorithmwasproposedbyLanczos[114]whoworkedonalgorithms for determiningeigenvalues of a matrix. His iterative algorithm yields the similarity transformation of a matrix into the tridiagonal form from which eigenvalues can be well approximated.Thethree-termrecurrencerelationofthe Lanczosprocedurecan be obtained by eliminating a vector from the conjugate direction algorithm scheme. Initially the conjugate gradient algorithm was called the Hestenes-Stiefel-Lanczos method [86].

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"The book describes important algorithms for the numerical treatment of unconstrained nonlinear optimization problems with many variables. ... This monograph is suitable as a text for a graduate course in computational optimization. It is useful to anyone active in this field. ... This book is well written and well organized. The argument is clear. Lists of algorithms as well as tables and figures facilitate for the reader the search for desired information in the text. The reference list is comprehensive and contains 214 items." (Sven-Ake Gustafson, Mathematical Reviews, Issue 2009 i)

"It is a very nice written book which can be used by researchers in optimization, in the teaching for seminars and by students ... . Lists of figures, tables and algorithms make this book to a useful compendium for research and teaching. A lot of bibliographical hints with respect to a large reference list make the reader known with the historical development of CG-methods ... . appendices with elements of topology, analysis, linear algebra and numerics of linear algebra make it to a self-contained book." (Armin Hoffmann, Zentralblatt MATH, Vol. 1171, 2009)

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Published: 20th November 2010

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Conjugate Direction Methods for Quadratic Problems	p. 1
Introduction	p. 1
Conjugate Direction Methods	p. 4
Conjugate Gradient Error Algorithms	p. 13
Conjugate Gradient Residual Algorithms	p. 21
The Method of Shortest Residuals	p. 26
Rate of Convergence	p. 28
Relations with a Direct Solver for a Set of Linear Equations	p. 38
Limited Memory Quasi-Newton Algorithms	p. 45
Reduced-Hessian Quasi-Newton Algorithms	p. 53
Limited Memory Reduced-Hessian Quasi-Newton Algorithms	p. 56
Conjugate Non-Gradient Algorithm	p. 59
Notes	p. 61
Conjugate Gradient Methods for Nonconvex Problems	p. 63
Introduction	p. 63
Line Search Methods	p. 65
General Convergence Results	p. 70
More-Thuente Step-Length Selection Algorithm	p. 76
Hager-Zhang Step-Length Selection Algorithm	p. 80
Nonlinear Conjugate Gradient Algorithms	p. 84
Global Convergence of the Fletcher-Reeves Algorithm	p. 87
Other Versions of the Fletcher-Reeves Algorithm	p. 96
Global Convergence of the Polak-Ribiere Algorithm	p. 98
Hestenes-Stiefel Versions of the Standard Conjugate Gradient Algorithm	p. 101
Notes	p. 106
Memoryless Quasi-Newton Methods	p. 109
Introduction	p. 109
Conjugate Gradient Algorithm as the Memoryless Quasi-Newton Method	p. 112
Beale's Restart Rule	p. 116
Convergence of the Shanno Algorithm	p. 118
Notes	p. 127
Preconditioned Conjugate Gradient Algorithms	p. 133
Introduction	p. 133
Rates of Convergence	p. 134
Conjugate Gradient Algorithm and the Newton Method	p. 139
Generic Preconditioned Conjugate Gradient Algorithm	p. 145
Quasi-Newton-like Variable Storage Conjugate Gradient Algorithm	p. 148
Scaling the Identity	p. 155
Notes	p. 158
Limited Memory Quasi-Newton Algorithms	p. 159
Introduction	p. 159
Global Convergence of the Limited Memory BFGS Algorithm	p. 163
Compact Representation of the BFGS Approximation of the Inverse Hessian Matrix	p. 169
The Compact Representation of the BFGS Approximation of the Hessian Matrix	p. 175
Numerical Experiments	p. 178
Notes	p. 186
The Method of Shortest Residuals and Nondifferentiable Optimization	p. 191
Introduction	p. 191
The Method of Conjugate Subgradient by Lemarechal and Wolfe	p. 192
Conjugate Subgradient Algorithm for Problems with Semismooth Functions	p. 204
Subgradient Algorithms with Finite Storage	p. 210
Notes	p. 215
The Method of Shortest Residuals for Differentiable Problems	p. 217
Introduction	p. 217
General Algorithm	p. 219
Versions of the Method of Shortest Residuals	p. 226
Polak-Ribiere Version of the Method of Shortest Residuals	p. 230
The Method of Shortest Residuals by Dai and Yuan	p. 233
Global Convergence of the Lemarechal-Wolfe Algorithm Without Restarts	p. 241
A Counter-Example	p. 244
Numerical Experiments: First Comparisons	p. 250
Numerical Experiments: Comparison to the Memoryless Quasi-Newton Method	p. 256
Numerical Experiments: Comparison to the Limited Memory Quasi-Newton Method	p. 257
Numerical Experiments: Comparison to the Hager-Zhang Algorithm	p. 264
Notes	p. 267
The Preconditioned Shortest Residuals Algorithm	p. 279
Introduction	p. 279
General Preconditioned Method of Shortest Residuals	p. 281
Globally Convergent Preconditioned Conjugate Gradient Algorithm	p. 286
Scaling Matrices	p. 288
Conjugate Gradient Algorithm with the BFGS Scaling Matrices	p. 291
Numerical Experiments	p. 293
Notes	p. 297
Optimization on a Polyhedron	p. 299
Introduction	p. 299
Exposed Sets	p. 308
The Identification of Active Constraints	p. 310
Gradient Projection Method	p. 313
On the Stopping Criterion	p. 319
Notes	p. 324
Conjugate Gradient Algorithms for Problems with Box Constraints	p. 327
Introduction	p. 327
General Convergence Theory	p. 328
The Globally Convergent Polak-Ribiere Version of Algorithm 10.1	p. 342
Convergence Analysis of the Fletcher-Reeves Version of Algorithm 10.1	p. 350
Numerical Experiments	p. 357
Notes	p. 359
Preconditioned Conjugate Gradient Algorithms for Problems with Box Constraints	p. 371
Introduction	p. 371
Active Constraints Identification by Solving Quadratic Problem	p. 371
The Preconditioned Shortest Residuals Algorithm: Outline	p. 376
The Preconditioned Shortest Residuals Algorithm: Global Convergence	p. 384
The Limited Memory Quasi-Newton Method for Problems with Box Constraints	p. 386
Numerical Algebra	p. 387
Algorithm 11.1 and Algorithm 11.2 Applied to Problems with Strongly Convex Functions	p. 389
Numerical Experiments	p. 390
Notes	p. 391
Preconditioned Conjugate Gradient Based Reduced-Hessian Methods	p. 399
Introduction	p. 399
BFGS Update with Column Scaling	p. 400
The Preconditioned Method of Shortest Residuals with Column Scaling	p. 409
Reduced-Hessian Quasi-Newton Algorithms	p. 412
Limited Memory Reduced-Hessian Quasi-Newton Algorithms	p. 420
Preconditioned Conjugate Gradient Based Reduced-Hessian Algorithm	p. 424
Notes	p. 428
Elements of Topology and Analysis	p. 429
The Topology of the Euclidean Space	p. 429
Continuity and Convexity	p. 432
Derivatives	p. 435
Elements of Linear Algebra	p. 441
Vector and Matrix Norms	p. 441
Spectral Decomposition	p. 442
Determinant and Trace	p. 447
Elements of Numerical Linear Algebra	p. 449
Condition Number and Linear Equations	p. 449
The LU and Cholesky Factorizations	p. 450
The QR Factorization	p. 452
Householder QR	p. 454
Givens QR	p. 456
Gram-Schmidt QR	p. 459
References	p. 463
Index	p. 473
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Conjugate Gradient Algorithms in Nonconvex Optimization

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