Table of Contents

Cover

Title

Copyright

Dedication

Preface

Chapter 1: Introduction

1.1 Problem Statement and Basic Definitions

1.2 Illustrative Examples

1.3 Guidelines for Model Construction

Exercises

Notes and References

Part 1: Convex Analysis

Chapter 2: Convex Sets

2.1 Convex Hulls

2.2 Closure and Interior of a Set

2.3 Weierstrass’s Theorem

2.4 Separation and Support of Sets

2.5 Convex Cones and Polarity

2.6 Polyhedral Sets, Extreme Points, and Extreme Directions

2.7 Linear Programming and the Simplex Method

Exercises

Notes and References

Chapter 3: Convex Functions and Generalizations

3.1 Definitions and Basic Properties

3.2 Subgradients of Convex Functions

3.3 Differentiable Convex Functions

3.4 Minima and Maxima of Convex Functions

3.5 Generalizations of Convex Functions

Exercises

Notes and References

Part 2: Optimality Conditions and Duality

Chapter 4: The Fritz John and Karush–Kuhn–Tucker Optimality Conditions

4.1 Unconstrained Problems

4.2 Problems Having Inequality Constraints

4.3 Problems Having Inequality and Equality Constraints

4.4 Second-Order Necessary and Sufficient Optimality Conditions for Constrained Problems

Exercises

Notes and References

Chapter 5: Constraint Qualifications

5.1 Cone of Tangents

5.2 Other Constraint Qualifications

5.3 Problems Having Inequality and Equality Constraints

Exercises

Notes and References

Chapter 6: Lagrangian Duality and Saddle Point Optimality Conditions

6.1 Lagrangian Dual Problem

6.2 Duality Theorems and Saddle Point Optimality Conditions

6.3 Properties of the Dual Function

6.4 Formulating and Solving the Dual Problem

6.5 Getting the Primal Solution

6.6 Linear and Quadratic Programs

Exercises

Notes and References

Part 3: Algorithms and Their Convergence

Chapter 7: The Concept of an Algorithm

7.1 Algorithms and Algorithmic Maps

7.2 Closed Maps and Convergence

7.3 Composition of Mappings

7.4 Comparison Among Algorithms

Exercises

Notes and References

Chapter 8: Unconstrained Optimization

8.1 Line Search Without Using Derivatives

8.2 Line Search Using Derivatives

8.3 Some Practical Line Search Methods

8.4 Closedness of the Line Search Algorithmic Map

8.5 Multidimensional Search Without Using Derivatives

8.6 Multidimensional Search Using Derivatives

8.7 Modification of Newton’s Method: Levenberg-Marquardt and Trust Region Methods

8.8 Methods Using Conjugate Directions: Quasi-Newton and Conjugate Gradient Methods

8.9 Subgradient Optimization Methods

Exercises

Notes and References

Chapter 9: Penalty and Barrier Functions

9.1 Concept of Penalty Functions

9.2 Exterior Penalty Function Methods

9.3 Exact Absolute Value and Augmented Lagrangian Penalty Methods

9.4 Barrier Function Methods

9.5 Polynomial-Time Interior Point Algorithms for Linear Programming Based on a Barrier Function

Exercises

Notes and References

Chapter 10: Methods of Feasible Directions

10.1 Method of Zoutendijk

10.2 Convergence Analysis of the Method of Zoutendijk

10.3 Successive Linear Programming Approach

10.4 Successive Quadratic Programming or Projected Lagrangian Approach

10.5 Gradient Projection Method of Rosen

10.6 Reduced Gradient Method of Wolfe and Generalized Reduced Gradient Method

10.7 Convex-Simplex Method of Zangwill

10.8 Effective First- and Second-Order Variants of the Reduced Gradient Method

Exercises

Notes and References

Chapter 11: Linear Complementary Problem, and Quadratic, Separable, Fractional, and Geometric Programming

11.1 Linear Complementary Problem

11.2 Convex and Nonconvex Quadratic Programming: Global Optimization Approaches

11.3 Separable Programming

11.4 Linear Fractional Programming

11.5 Geometric Programming

Exercises

Notes and References

Appendix A: Mathematical Review

Appendix B: Summary of Convexity, Optimality Conditions, and Duality

Bibliography

Index

End User License Agreement

List of Tables

Chapter 6: Lagrangian Duality and Saddle Point Optimality Conditions

Table 6.1 Summary of Computations for Example 6.4.1

Chapter 8: Unconstrained Optimization

Table 8.1 Summary of Computations for the Golden Section Method

Table 8.2 Summary of Computations for the Fibonacci Search Method

Table 8.3 Summary of Computations for the Bisection Search Method

Table 8.4 Summary of Computations for Newton’s Method Starting from λ₁ = 0.4

Table 8.5 Summary of Computations for Newton’s Method Starting from λ₁ = 0.6

Table 8.6 Summary of Computations for the Cyclic Coordinate Method

Table 8.7 Summary of Computations for the Method of Hooke and Jeeves Using Line Searches

Table 8.8 Summary of Computations for the Method of Hooke and Jeeves with Discrete Steps

Table 8.9 Summary of Computations for the Method of Rosenbrock Using Line Searches

Table 8.10 Summary of Computations for Rosenbrock’s Method Using Discrete Steps

Table 8.11 Summary of Computations for the Method of Steepest Descent

Table 8.12 Summary of Computations for the Method of Newton

Table 8.13 Summary of Computations for the Davidon–Fletcher–Powell Method

Table 8.14 Summary of Computations for the Method of Fletcher and Reeves

Chapter 9: Penalty and Barrier Functions

Table 9.1 Summary of Computations for the Penalty Function Method

Table 9.2 Summary of Computations for the Barrier Function Method

Chapter 10: Methods of Feasible Directions

Table 10.1 summarizes the computations for solving the problem. The progress of the algorithm is shown in Figure 10.5.

Table 10.2 Summary of Computations for the Method of Zoutendijk for the Case of Nonlinear Constraints

Table 10.3 Summary of the Topkis–Veinott Method

Table 10.4 Summary of Computations for the Gradient Project Method of Rosen

Table 10.5 Summary of Computations for the Reduced Gradient Method of Wolfe

Table 10.6 Summary of Computation for the Convex-Simplex Method of Zangwill

List of Illustrations

Chapter 1: Introduction

Figure 1.1 Geometric solution of a nonlinear problem.

Figure 1.2 Discrete control system.

Figure 1.3 Two-bar truss.

Figure 1.4 Journal bearing assembly.

Figure 1.5 Typical electrical devices in a circuit.

Figure 1.6 Typical river basin.

Figure 1.7 Utility functions.

Chapter 2: Convex Sets

Figure 2.1 Convex and nonconvex sets.

Figure 2.2 Convex hulls.

Figure 2.3 Polytope and simplex.

Figure 2.4 Line segment joining points in the closure and interior of a set.

Figure 2.5 Nonexistence of a minimizing solution.

Figure 2.6 Parallelogram law.

Figure 2.7 Minimum distance to a closed convex set.

Figure 2.8 Hyperplane and corresponding half-spaces.

Figure 2.9 Various types of separation.

Figure 2.10 Farkas’s theorem.

Figure 2.11 Supporting hyperplanes.

Figure 2.12 Supporting hyperplane.

Figure 2.13 Nonexistence of a strongly separating hyperplane.

Figure 2.14 Cones.

Figure 2.15 Polar cones.

Figure 2.16 Polyhedral set.

Figure 2.17 Extreme points.

Figure 2.18 Extreme directions.

Figure 2.19 Linear programming example.

Chapter 3: Convex Functions and Generalizations

Figure 3.1 Convex and concave functions.

Figure 3.2 Nonexistence of the directional derivative of f at images in the direction d.

Figure 3.3 Epigraphs and hypographs.

Figure 3.4 Geometric interpretation of subgradients.

Figure 3.5 Setup for Example 3.2.4.

Figure 3.6 Local and global minima.

Figure 3.7 Strict local minima are not necessarily strong local minima.

Figure 3.8 Geometry for Theorems 3.4.3 and 3.4.4.

Figure 3.9 Setup for Example 3.4.5.

Figure 3.10 Quasiconvex and quasiconcave functions: (a) quasiconvex, (b) quasiconcave, (c) neither quasiconvex nor quasiconcave, (d) quasimonotone.

Figure 3.11 Strictly quasiconvex and strictly quasiconcave functions: (a) strictly quasiconvex, (b) strictly quasiconvex, (c) strictly quasiconcave, (d) neither strictly quasiconvex nor quasiconcave.

Figure 3.12 Pseudoconvex and pseudoconcave functions: (a) pseudoconvex, (b) both pseudoconvex and pseudoconcave, (c) neither pseudoconvex nor pseudoconcave.

Figure 3.13 Relationship among various types of convexity.

Figure 3.14 Various types of convexity at a point, (a) Convexity and strict convexity: f is convex but not strictly convex at x₁; f is both convex and strictly convex at x₂. (b) Pseudoconvexity and strict pseudoconvexity: f is pseudoconvex but not strictly pseudoconvex at x₁; f is both pseudoconvex and strictly pseudoconvex at x₂. (c) Quasiconvexity, strict quasiconvexity, and strong quasiconvexity: f is quasiconvex but neither strictly quasiconvex nor strongly quasiconvex at x₁; f is both quasiconvex and strictly quasiconvex at x₂ but not strongly quasiconvex at x₂; f is quasiconvex, strictly quasiconvex, and strongly quasiconvex at x₃.

Chapter 4: The Fritz John and Karush–Kuhn–Tucker Optimality Conditions

Figure 4.1 Regions of zero, positive, and negative values of images.

Figure 4.2 Saddle point at images.

Figure 4.3 Necessary condition images.

Figure 4.4 images at a nonoptimal point.

Figure 4.5 images at an optimal point.

Figure 4.6 Feasible Region in Example 4.2.9.

Figure 4.7 Feasible region in Example 4.2.10.

Figure 4.8 Feasible Region in Example 4.2.11.

Figure 4.9 FJ conditions are not sufficient for optimality for LP problems.

Figure 4.10 Geometric illustration of the KKT conditions.

Figure 4.11 KKT conditions are not necessary for convex programming problems.

Figure 4.12 Setup for Example 4.3.5.

Figure 4.13 Caution on deleting nonbinding constraints for nonconvex problems.

Figure 4.14 Two cases of optimal solutions: Example 4.4.4.

Chapter 5: Constraint Qualifications

Figure 5.1 Cone of tangents.

Figure 5.2 Relationships among various constraint qualifications for inequality-constrained problems.

Figure 5.3 Relationships among constraint qualifications for problems having inequality and equality constraints.

Chapter 6: Lagrangian Duality and Saddle Point Optimality Conditions

Figure 6.1 Geometric interpretation of Lagrangian duality.

Figure 6.2 Geometric interpretation of Example 6.1.1.

Figure 6.3 Duality gap.

Figure 6.4 Dual function for Example 6.2.2.

Figure 6.5 Geometric interpretation of Example 6.2.2.

Figure 6.6 Solution to Example 6.2.8.

Figure 6.7 Subgradients.

Figure 6.8 Cone of ascent directions in Example 6.3.8.

Figure 6.9 Tangential approximation of θ.

Chapter 7: The Concept of an Algorithm

Figure 7.1 Algorithmic maps.

Figure 7.2 Nonconvergent algorithmic map.

Figure 7.3 Composite maps.

Chapter 8: Unconstrained Optimization

Figure 8.1 Reducing the interval of uncertainty.

Figure 8.2 Uniform search

Figure 8.3 Possible intervals of uncertainty.

Figure 8.4 Golden section rule.

Figure 8.5 Quadratic-fit line search.

Figure 8.6 Armijo’s rule.

Figure 8.7 Cyclic coordinate method.

Figure 8.8 Effect of a sharp-edged valley.

Figure 8.9 Method of Hooke and Jeeves.

Figure 8.10 Method of Hooke and Jeeves using line searches. Method of Hooke and Jeeves with Discrete Steps

Figure 8.11 Method of Hooke and Jeeves using discrete steps starting from (0.0,3.0). (The numbers denote the order in which points are generated.)

Figure 8.12 Method of Hooke and Jeeves using line searches. (The numbers denote the order in which points are generated.)

Figure 8.13 Rosenbrock’s procedure using discrete steps.

Figure 8.14 Method of Rosenbrock using line searches.

Figure 8.15 Rosenbrock’s procedure using discrete steps. (The numbers denote the order in which points are generated.)

Figure 8.16 Method of steepest descent.

Figure 8.17 Convergence rate analysis of the steepest descent algorithm.

Figure 8.18 Method of Newton.

Figure 8.19 Dog-leg trajectory.

Figure 8.20 Illustration of conjugate directions.

Figure 8.21 Expanding subspace property.

Figure 8.22 Davidon–Fletcher–Powell method.

Figure 8.23 Method of Fletcher and Reeves

Figure 8.24 Equivalence between the conjugate gradient method and PARTAN

Figure 8.25 Contours of/, in Example 8.9.1.

Chapter 9: Penalty and Barrier Functions

Figure 9.1 Penalty and auxiliary functions.

Figure 9.2 Geometry of penalty functions in the (h, f) space.

Figure 9.3 Penalty functions and nonconvex problems.

Figure 9.4 III-conditioning effect of a large μ value.

Figure 9.5 Penalty function method.

Figure 9.6 Geometric interpretation of the absolute penalty function.

Figure 9.7 Geometric interpretation of the augmented Lagrangian penalty function.

Figure 9.8 Barrier and auxiliary functions.

Figure 9.9 Barrier function method.

Chapter 10: Methods of Feasible Directions

Figure 10.1 Improving feasible directions.

Figure 10.2 Iteration 1.

Figure 10.3 Iteration 2.

Figure 10.4 Termination at Iteration 3.

Figure 10.5 Method of Zoutendijk for the case of linear constraints.

Figure 10.6 Improving feasible directions for nonlinear constraints.

Figure 10.7 Method of Zoutendijk for the case of nonlinear inequality constraints.

Figure 10.8 Nonlinear equality constraints.

Figure 10.9 Effect of near-binding constraints.

Figure 10.10 The direction-finding map D is not closed.

Figure 10.11 The line search map M is not closed.

Figure 10.12 Method of Topkis and Veinott.

Figure 10.13 Solution to Example 10.3.2.

Figure 10.14 Solution of subproblem QP.

Figure 10.15 Projecting the gradient.

Figure 10.16 Gradient projection method of Rosen

Figure 10.17 Projecting the gradient in the presence of nonlinear constraints.

Figure 10.18 The direction-finding map is not closed.

Figure 10.19 Illustration of the reduced gradient method of Wolfe.

Figure 10.20 Convex-simplex method of Zangwill.

Figure 10.21 Zigzagging of the convex-simplex method.

Chapter 11: Linear Complementary Problem, and Quadratic, Separable, Fractional, and Geometric Programming

Figure 11.1 Points generated by the complementary pivoting algorithm.

Figure 11.2 Unbounded optimal solution and ray termination.

Figure 11.3 Piecewise linear approximation of a function.

Figure 11.4 Importance of adjacency in approximation.

Figure 11.5 Feasible region for Example 11.4.3.

Appendix B: Summary of Convexity, Optimality Conditions, and Duality

Figure B.l Convexity.

Figure B.2 Separating and supporting hyperplanes.

Figure B.3 Convex and concave functions.

Figure B.4 Quasiconvexity and pseudoconvexity.

Figure B.5 Relationship among various types of convexity.

Figure B.6 The KKT conditions.

Guide

Cover

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NONLINEAR PROGRAMMING

Theory and Algorithms

Third Edition

MOKHTAR S. BAZARAA

Georgia Institute of Technology
School of Industrial and Systems Engineering
Atlanta, Georgia

HANIF D. SHERALI

Virginia Polytechnic Institute and State University
Grado Department of Industrial and Systems Engineering
Blacksburg, Virginia

C. M. SHETTY

Georgia Institute of Technology
School of Industrial and Systems Engineering
Atlanta, Georgia

Copyright © 2006 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.

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Library of Congress Cataloging-in-Publication Data:

Bazaraa, M. S.

Nonlinear programming : theory and algorithms / Mokhtar S. Bazaraa, Hanif D. Sherali, C. M. Shetty.—3rd ed.

p. cm.

“Wiley-Interscience.”

Includes bibliographical references and index.

ISBN-13: 978-0-471-48600-8 (cloth: alk. paper)

ISBN-10: 0-471-48600-0 (cloth: alk. paper)

1. Nonlinear programming. I. Sherali, Hanif D., 1952–. II. Shetty, C. M, 1929–. III Title.

T57.8.B39 2006

519.7′6—dc22

2005054230

10 9 8 7 6 5 4 3 2 1

Dedicated to our parents

Preface

Nonlinear programming deals with the problem of optimizing an objective function in the presence of equality and inequality constraints. If all the functions are linear, we obviously have a linear program. Otherwise, the problem is called a nonlinear program. The development of highly efficient and robust algorithms and software for linear programming, the advent of high-speed computers, and the education of managers and practitioners in regard to the advantages and profitability of mathematical modeling and analysis have made linear programming an important tool for solving problems in diverse fields. However, many realistic problems cannot be adequately represented or approximated as a linear program, owing to the nature of the nonlinearity of the objective function and/or the nonlinearity of any of the constraints. Efforts to solve such nonlinear problems efficiently have made rapid progress during the past four decades. This book presents these developments in a logical and self-contained form.

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