This research was supported by the World Bank and the Social Sciences and Humanities R~search Council of Canada. Neither institution is responsible for the views expressed in this paper. The author is indebted to V. Corbo, A. Haymer, Y. Kanemoto, K. Lee, K. Mera, Hal Varian and A. Walters for helpful comments and to Elizabeth Lambert, Shehnaz Motani', and Jeanette Leigh Paisley for excellent typing services. I would like to dedicate this book to my wife, Virginia. TABLE OF CONTENTS 1. Introduction . . . •. . ••. . •. ••. •••. . . . •. . . . . •. •. . . •. . . •. •. •. . •. . . •. ••. • 2. A Simple Producer Benefit Measure . . ••. •••. ••••. •••••••••••. ••••••. • 8 3. Willingness to P~v Functions and Marginal Cost Functions ••••••••••• 15 4. Approximate Benefit Measures 30 5. Problems with the Producer Benefit Measure ••••••••••••• ! ••••••••••• 41 5. 1. Static versus Dynamic Benefit Measures ••••••••••••••••••• 41 5. 2. The Problem of Endogenous Prices for Local Goons ••••••••• 48 5. 3. The Neglect of Consumer Benefits' ••••••••••••••••••••••••• 56 6. Alternative Approaches to Benefit Measurement •••••••••••••••••••••• 70 6. 1. The Questionnaire or Sample Survey Approach •••••••••••••• 70 6. 2. Ex Post Accounting Approaches •••••••••••••••••••••••••••• 72 6. 3. Engineering and Mathematical Programming Approaches •••••• 74 6. 4. The Applied General Equilibrium Modelling Approach ~ •••••• 75 6. 5. The Differential Approach •••••••••••••••••••••••••••••••• 77 6. 6. The Econometric Approach . . •••••••••. •. ••••••. •••••••••. •• 79 7.
This classic volume covers the fundamentals of two closely related topics: linear systems (linear equations and least-squares) and linear programming (optimizing a linear function subject to linear constraints). For each problem class, stable and efficient numerical algorithms intended for a finite-precision environment are derived and analyzed. While linear algebra and optimization have made huge advances since this book first appeared in 1991, the fundamental principles have not changed. These topics were rarely taught with a unified perspective, and, somewhat surprisingly, this remains true 30 years later. As a result, some of the material in this book can be difficult to find elsewhere—in particular, techniques for updating the LU factorization, descriptions of the simplex method applied to all-inequality form, and the analysis of what happens when using an approximate inverse to solve Ax=b. Numerical Linear Algebra and Optimization is primarily a reference for students who want to learn about numerical techniques for solving linear systems and/or linear programming using the simplex method; however, Chapters 6, 7, and 8 can be used as the text for an upper-division course on linear least squares and linear programming. Understanding is enhanced by numerous exercises.
In the intervening years since this book was published in 1981, the field of optimization has been exceptionally lively. This fertility has involved not only progress in theory, but also faster numerical algorithms and extensions into unexpected or previously unknown areas such as semidefinite programming. Despite these changes, many of the important principles and much of the intuition can be found in this Classics version of Practical Optimization. This book provides model algorithms and pseudocode, useful tools for users who prefer to write their own code as well as for those who want to understand externally provided code. It presents algorithms in a step-by-step format, revealing the overall structure of the underlying procedures and thereby allowing a high-level perspective on the fundamental differences. And it contains a wealth of techniques and strategies that are well suited for optimization in the twenty-first century, and particularly in the now-flourishing fields of data science, “big data,” and machine learning. Practical Optimization is appropriate for advanced undergraduates, graduate students, and researchers interested in methods for solving optimization problems.
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