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Nonlinear Programming

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By Olvi L. Mangasarian


1994 / xv + 220 pages / Softcover / ISBN: 978-0-898713-41-1 / List Price $61.50 / SIAM Member Price $43.05 / Order Code CL10

This reprint of the 1969 book of the same name is a concise, rigorous, yet accessible, account of the fundamentals of constrained optimization theory. Many problems arising in diverse fields such as machine learning, medicine, chemical engineering, structural design, and airline scheduling can be reduced to a constrained optimization problem. This book provides readers with the fundamentals needed to study and solve such problems.

Beginning with a chapter on linear inequalities and theorems of the alternative, basics of convex sets and separation theorems are then derived based on these theorems. This is followed by a chapter on convex functions that includes theorems of the alternative for such functions. These results are used in obtaining the saddlepoint optimality conditions of nonlinear programming without differentiability assumptions. Properties of differentiable convex functions are derived and then used in two key chapters of the book, one on optimality conditions for differentiable nonlinear programs and one on duality in nonlinear programming. Generalizations of convex functions to pseudoconvex and quasiconvex functions are given and then used to obtain generalized optimality conditions and duality results in the presence of nonlinear equality constraints.

The book has four useful self-contained appendices on vectors and matrices, topological properties of n-dimensional real space, continuity and minimization, and differentiable functions.

Undergraduates with an advanced calculus background and graduate students in computer science, industrial engineering, operations research, electrical engineering, economics, business, mathematics, and civil and mechanical engineering will find this book of great use. It will also be of interest to researchers in oil, investment, chemical, and software companies, as well as banks and airlines.

Preface to the Classic Edition; Chapter 1: The Nonlinear Programming Problem, Preliminary Concepts, and Notation; Chapter 2: Linear Inequalities and Theorems of the Alternative; Chapter 3: Convex Sets in Rn; Chapter 4: Convex and Concave Functions; Chapter 5: Saddlepoint Optimality Criteria in Nonlinear Programming Without Differentiability; Chapter 6: Differentiable Convex and Concave Functions; Chapter 7: Optimality Criteria in Nonlinear Programming with Differentiability; Chapter 8: Duality in Nonlinear Programming; Chapter 9: Generalizations of Convex Functions: Quasiconvex, Strictly Quasiconvex, and Pseudoconvex Functions; Chapter 10: Optimality and Duality for Generalized Convex and Concave Functions; Chapter 11: Optimality and Duality in the Presence of Nonlinear Equality Constraints; Appendix A: Vectors and Matrices; Appendix B: Resume of Some Topological Properties of Rn; Appendix C: Continuous and Semicontinuous Functions, Minima and Infima; Appendix D: Differentiable Functions, Mean-value and Implicit Function Theorems; Bibliography; Name Index; Subject Index.

ISBN: 9780898713411

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