The Mathematics of Nonlinear Programming

by Anthony L. Peressini, Francis E. Sullivan, J.J. Jr. Uhl

Overview

Nonlinear programming provides an excellent opportunity to explore an interesting variety of pure and solidly applicable mathematics, numerical analysis, and computing. This text develops some of the ideas and techniques involved in the optimization methods using calculus, leading to the study of convexity. This is followed by material on basic numerical methods, least squares, the Karush-Kuhn-Tucker theorem, penalty functions, and Lagrange multipliers. The authors have aimed their presentation at the student who has a working knowledge of matrix algebra and advanced calculus, but has had no previous exposure to optimization.

Product Details

ISBN-13:	9781461269892
Publisher:	Springer New York
Publication date:	09/30/2012
Series:	Undergraduate Texts in Mathematics
Edition description:	Softcover reprint of the original 1st ed. 1988
Pages:	276
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

1 Unconstrained Optimization via Calculus.- 1.1. Functions of One Variable.- 1.2. Functions of Several Variables.- 1.3. Positive and Negative Definite Matrices and Optimization.- 1.4. Coercive Functions and Global Minimizers.- 1.5. Eigenvalues and Positive Definite Matrices.- Exercises.- 2 Convex Sets and Convex Functions.- 2.1. Convex Sets.- 2.2. Some Illustrations of Convex Sets in Economics— Linear Production Models.- 2.3. Convex Functions.- 2.4. Convexity and the Arithmetic-Geometric Mean Inequality— An Introduction to Geometric Programming.- 2.5. Unconstrained Geometric Programming.- 2.6. Convexity and Other Inequalities.- Exercises.- 3 Iterative Methods for Unconstrained Optimization.- 3.1. Newton’s Method.- 3.2. The Method of Steepest Descent.- 3.3. Beyond Steepest Descent.- 3.4. Broyden’s Method.- 3.5. Secant Methods for Minimization.- Exercises.- 4 Least Squares Optimization.- 4.1. Least Squares Fit.- 4.2. Subspaces and Projections.- 4.3. Minimum Norm Solutions of Underdetermined Linear Systems.- 4.4. Generalized Inner Products and Norms; The Portfolio Problem.- Exercises.- 5 Convex Programming and the Karush-Kuhn-Tucker Conditions.- 5.1. Separation and Support Theorems for Convex Sets.- 5.2. Convex Programming; The Karush-Kuhn-Tucker Theorem.- 5.3. The Karush-Kuhn-Tucker Theorem and Constrained Geometric Programming.- 5.4. Dual Convex Programs.- 5.5. Trust Regions.- Exercises.- 6 Penalty Methods.- 6.1. Penalty Functions.- 6.2. The Penalty Method.- 6.3. Applications of the Penalty Function Method to Convex Programs.- Exercises.- 7 Optimization with Equality Constraints.- 7.1. Surfaces and Their Tangent Planes.- 7.2. Lagrange Multipliers and the Karush-Kuhn-Tucker Theorem for Mixed Constraints.- 7.3. Quadratic Programming.- Exercises.