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Numerical Optimization (Second Edition).pdf
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Springer Series in Operations Research
and Financial Engineering
Editors:
Thomas V. Mikosch Sidney I. Resnick
Stephen M. Robinson
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Springer Series in Operation Research and
Financial Engineering
Altiok: Performance Analysis of Manufacturing Systems
Birge and Louveaux: Introduction to Stochastic Programming
Bonnans and Shapiro: Perturbation Analysis of Optimization Problems
Bramel, Chen, and Simchi-Levi: The Logic of Logistics: Theory, Algorithms,
and Applications for Logistics and Supply Chain Management
(second edition)
Dantzig and Thapa: Linear Programming 1: Introduction
Dantzig and Thapa: Linear Programming 2: Theory and Extensions
Drezner (Editor): Facility Location: A Survey of Applications and Methods
Facchinei and Pang: Finite-Dimensional Variational Inequalities and
Complementarity Problems, Volume I
Facchinei and Pang: Finite-Dimensional Variational Inequalities and
Complementarity Problems, Volume II
Fishman: Discrete-Event Simulation: Modeling, Programming, and Analysis
Fishman: Monte Carlo: Concepts, Algorithms, and Applications
Haas: Stochastic Petri Nets: Modeling, Stability, Simulation
Klamroth: Single-Facility Location Problems with Barriers
Muckstadt: Analysis and Algorithms for Service Parts Supply Chains
Nocedal and Wright: Numerical Optimization
Olson: Decision Aids for Selection Problems
Pinedo: Planning and Scheduling in Manufacturing and Services
Pochet and Wolsey: Production Planning by Mixed Integer Programming
Whitt: Stochastic-Process Limits: An Introduction to Stochastic-Process
Limits and Their Application to Queues
Yao (Editor): Stochastic Modeling and Analysis of Manufacturing Systems
Yao and Zheng: Dynamic Control of Quality in Production-Inventory
Systems: Coordination and Optimization
Yeung and Petrosyan: Cooperative Stochastic Differential Games
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Jorge Nocedal
Stephen J. Wright
Numerical Optimization
Second Edition
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Stephen J. Wright
Computer Sciences Department
University of Wisconsin
1210 West Dayton Street
Madison, WI 53706–1613
USA
swright@cs.wisc.edu
Stephen M. Robinson
Department of Industrial and Systems
Engineering
University of Wisconsin
1513 University Avenue
Madison, WI 53706–1539
USA
smrobins@facstaff.wise.edu
Jorge Nocedal
EECS Department
Northwestern University
Evanston, IL 60208-3118
USA
nocedal@eecs.northwestern.edu
Series Editors:
Thomas V. Mikosch
University of Copenhagen
Laboratory of Actuarial Mathematics
DK-1017 Copenhagen
Denmark
mikosch@act.ku.dk
Sidney I. Resnick
Cornell University
School of Operations Research and
Industrial Engineering
Ithaca, NY 14853
USA
sirl@cornell.edu
Mathematics Subject Classification (2000): 90B30, 90C11, 90-01, 90-02
Library of Congress Control Number: 2006923897
ISBN-10: 0-387-30303-0
ISBN-13: 978-0387-30303-1
Printed on acid-free paper.
C 2006 Springer Science+Business Media, LLC.
All rights reserved. This work may not be translated or copied in whole or in part without the written permission
of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for
brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not
identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary
rights.
Printed in the United States of America.
(TB/HAM)
9 8 7 6 5 4 3 2 1
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To Sue, Isabel and Martin
and
To Mum and Dad
Contents
Preface
Preface to the Second Edition
1
Introduction
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Mathematical Formulation .
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Example: A Transportation Problem .
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Continuous versus Discrete Optimization .
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Constrained and Unconstrained Optimization .
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Global and Local Optimization .
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Stochastic and Deterministic Optimization .
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Convexity
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Optimization Algorithms
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Notes and References
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2 Fundamentals of Unconstrained Optimization
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2.1 What Is a Solution?
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C O N T E N T S
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Recognizing a Local Minimum .
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Nonsmooth Problems .
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Overview of Algorithms .
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Two Strategies: Line Search and Trust Region .
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Search Directions for Line Search Methods .
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Models for Trust-Region Methods .
Scaling .
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Step Length .
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The Wolfe Conditions .
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The Goldstein Conditions .
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Sufficient Decrease and Backtracking .
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Convergence of Line Search Methods .
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Rate of Convergence .
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Convergence Rate of Steepest Descent .
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Newton’s Method .
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Newton’s Method with Hessian Modification .
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Eigenvalue Modification .
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Adding a Multiple of the Identity .
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Modified Cholesky Factorization .
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Modified Symmetric Indefinite Factorization .
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Step-Length Selection Algorithms .
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Interpolation .
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Initial Step Length .
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A Line Search Algorithm for the Wolfe Conditions .
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Notes and References
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4 Trust-Region Methods
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Outline of the Trust-Region Approach .
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Algorithms Based on the Cauchy Point
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The Cauchy Point
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Improving on the Cauchy Point .
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Two-Dimensional Subspace Minimization .
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Global Convergence .
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Reduction Obtained by the Cauchy Point .
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Convergence to Stationary Points .
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