Numerical Optimization: theoretical and practical aspects : with 26 figures

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Springer Science & Business Media, 2003 - Mathematics - 419 pages
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Starting with illustrative real-world examples, this book exposes in a tutorial way algorithms for numerical optimization: fundamental ones (Newtonian methods, line-searches, trust-region, sequential quadratic programming, etc.), as well as more specialized and advanced ones (nonsmooth optimization, decomposition techniques, and interior-point methods). Most of these algorithms are explained in a detailed manner, allowing straightforward implementation. Theoretical aspects are addressed with care, often using minimal assumptions. The present version contains substantial changes with respect to the first edition. Part I on unconstrained optimization has been completed with a section on quadratic programming. Part II on nonsmooth optimization has been thoroughly reorganized and expanded. In addition, nontrivial application problems have been inserted, in the form of computational exercises. These should help the reader to get a better understanding of optimization methods beyond their abstract description, by addressing important features to be taken into account when passing to implementation of any numerical algorithm. This level of detail is intended to familiarize the reader with some of the crucial questions of numerical optimization: how algorithms operate, why they converge, difficulties that may be encountered and their possible remedies.
 

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Contents

General Introduction
3
Basic Methods
25
LineSearches
37
Newtonian Methods
51
Conjugate Gradient
67
Special Methods
77
Some Theory of Nonsmooth Optimization
95
Some Methods in Nonsmooth Optimization
101
Globalization by LineSearch
235
QuasiNewton Versions
265
InteriorPoint Algorithms for Linear and Quadratic
283
Linearly Constrained Optimization and Simplex Algorithm291
291
Linear Monotone Complementarity and Associated
307
PredictorCorrector Algorithms
329
NonFeasible Algorithms
345
SelfDuality
357

Bundle Methods The Quest of Descent
117
Decomposition and Duality
137
Newtons Methods in Constrained Optimization
149
Background
157
Local Methods for Problems with Equality Constraints
169
Local Methods for Problems with Equality and Inequality
203
Exact Penalization
217
OneStep Methods
367
Complexity of Linear Optimization Problems with
383
Karmarkars Algorithm
389
References
397
Index
415
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Page 398 - RH Byrd. ME Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming.
Page 402 - PE Gill, W. Murray, and MA Saunders, SNOPT: An SQP algorithm for large-scale constrained optimization, Numerical Analysis Report 97-2, Department of Mathematics, University of California, San Diego, La Jolla, CA, (1997).

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