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This book is an introduction to calculus and linear algebra for students of disciplines such as economics, finance, business, management, and accounting. It is intended for readers who may have already encountered some differential calculus, and it will also be appropriate for those with less experience, possibly used in conjunction with one of the many more elementary texts on basic mathematics. Parts of this book arise from a lecture course given by the authors to students of economics, management, accounting and finance, and management sciences at the London School of Economics. We thank Duncan Anthony, Reza Arabsheibani, Juliet Biggs and, particularly, Graham Brightwell for their invaluable comments on various drafts of the book. The final draft was read by Dr Stephen Siklos of Cambridge University, and his pertinent comments resulted in a number of improvements. We are also grateful to Roger Astley of Cambridge University Press for his efficient handling of the project, and to Alison Adcock for her help in preparing the manuscript. London, October 1995.
Contents xi Preface Chapter I Introduction I I General Remarks 1.2 Notation and Mathematical Background 1.3 Unconstrained Minimization 1.3.l Convergence Analysis of Gradient Methods 1.3.2 Steepest Descent and Scaling 1.3.3 Newton's Method and Its Modifications 1.3.4 Conjugate Direction and Conjugate Gradient Methods 1.3.5 Quasi-Newton Methods 1.3.6 Methods Not Requiring Evaluation of Derivatives 1.4 Constrained Minimization 1.5 Algorithms for Minimization Subject to Simple Constraints 1.6 Notes and Sources I 18 20 39 40 49 59 65 66 76 93 Chapter The Method of Multipliers for Equality Constrained Problems 2.1 The Quadratic Penalty Function Method 2.2 The Original Method of Multipliers 2.2.1 Geometric Interpretation 2.2.2 Existence of Local Minima of the Augmented Lagrangian 2.2.3 The Primal Functional 2.2.4 Convergence Analysis 2.2.5 Comparison with the Penalty Method�Computational Aspects 2.3 Duality Framework for the Method of Multipliers 2.3.l Stepsize Analysis for the Method of Multipliers 2.3.2 The Second-Order Multiplier Iteration 2.3.3 Quasi-Newton Versions of the Second-Order Iteration 2.3.4 Geometric Interpretation of the Second-Order Multiplier Iteration 96 104 105 107 113 115 121 125 126 133 138 139 vii