【资料名称】：An Introduction to Continuous Optimization
【资料作者】：Niclas Andreasson, Anton Evgrafov, M. Patriksson
【出版社】：    Studentlitteratur AB
【简介及目录】：   

Product details  

• Paperback: 400 pages
• Publisher: Studentlitteratur AB,Sweden (8 May 2006)
• Language English
• ISBN-10: 9144044550

Product Description  
Optimisation, or mathematical programming, is a fundamentalsubjectwithin decision science and operations research, in whichmathematicaldecision models are constructed, analysed, and solved. Thisbook'sfocus lies on providing a basis for the analysis of optimisationmodelsand of candidate optimal solutions, especially forcontinuousoptimisation models. Themain part of the mathematical materialtherefore concerns the analysisand linear algebra that underlie theworkings of convexity and duality,and necessary/sufficientlocal/global optimality conditions forunconstrained and constrainedoptimisation problems. Natural algorithmsare then developed from theseoptimality conditions, and their mostimportant convergencecharacteristics are analysed. This book answersmany more questions ofthe form: 'Why/why not?' than 'How?'.This choiceof focus is incontrast to books mainly providing numerical guidelinesas to howoptimisation problems should be solved. We use onlyelementarymathematics in the development of the book, yet arerigorousthroughout. This book provides lecture, exercise and readingmaterialfor a first course on continuous optimisation andmathematicalprogramming, geared towards third-year students, and hasalready beenused as such, in the form of lecture notes, for nearly tenyears. Thisbook can be used in optimisation courses at any engineeringdepartmentas well as in mathematics, economics, and business schools.It is aperfect starting book for anyone who wishes to develophis/herunderstanding of the subject of optimisation, before actuallyapplyingit.

Table of Contents
I Introduction 1
1 Modelling and classification 3
1.1 Modelling of optimization problems . . . . . . . . . . . . . 3
1.2 A quick glance at optimization history . . . . . . . . . . . 9
1.3 Classification of optimization models . . . . . . . . . . . . 11
1.4 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Applications and modelling examples . . . . . . . . . . . . 15
1.6 Defining the field . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 On optimality conditions . . . . . . . . . . . . . . . . . . . 16
1.8 Soft and hard constraints . . . . . . . . . . . . . . . . . . 18
1.8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . 18
1.8.2 A derivation of the exterior penalty function . . . 19
1.9 A road map through the material . . . . . . . . . . . . . . 20
1.10 On the background of this book and a didactics statement 25
1.11 Illustrating the theory . . . . . . . . . . . . . . . . . . . . 26
1.12 Notes and further reading . . . . . . . . . . . . . . . . . . 27
1.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
II Fundamentals 31
2 Analysis and algebra—A summary 33
2.1 Reductio ad absurdum . . . . . . . . . . . . . . . . . . . . 33
2.2 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Convex analysis 41
3.1 Convexity of sets . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Polyhedral theory . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Convex hulls . . . . . . . . . . . . . . . . . . . . . 423.2.2 Polytopes . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.4 The Separation Theorem and Farkas’ Lemma . . . 52
3.3 Convex functions . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Application: the projection of a vector onto a convex set . 66
3.5 Notes and further reading . . . . . . . . . . . . . . . . . . 69
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
III Optimality Conditions 73
4 An introduction to optimality conditions 75
4.1 Local and global optimality . . . . . . . . . . . . . . . . . 75
4.2 Existence of optimal solutions . . . . . . . . . . . . . . . . 78
4.2.1 A classic result . . . . . . . . . . . . . . . . . . . . 78
4.2.2 ∗Non-standard results . . . . . . . . . . . . . . . . 81
4.2.3 Special optimal solution sets . . . . . . . . . . . . 83
4.3 Optimality in unconstrained optimization . . . . . . . . . 84
4.4 Optimality for optimization over convex sets . . . . . . . . 88
4.5 Near-optimality in convex optimization . . . . . . . . . . . 95
4.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.6.1 Continuity of convex functions . . . . . . . . . . . 96
4.6.2 The Separation Theorem . . . . . . . . . . . . . . 98
4.6.3 Euclidean projection . . . . . . . . . . . . . . . . . 99
4.6.4 Fixed point theorems . . . . . . . . . . . . . . . . 100
4.7 Notes and further reading . . . . . . . . . . . . . . . . . . 106
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5 Optimality conditions 111
5.1 Relations between optimality conditions and CQs at a glance111
5.2 A note of caution . . . . . . . . . . . . . . . . . . . . . . . 112
5.3 Geometric optimality conditions . . . . . . . . . . . . . . 114
5.4 The Fritz John conditions . . . . . . . . . . . . . . . . . . 118
5.5 The Karush–Kuhn–Tucker conditions . . . . . . . . . . . . 124
5.6 Proper treatment of equality constraints . . . . . . . . . . 128
5.7 Constraint qualifications . . . . . . . . . . . . . . . . . . . 130
5.7.1 Mangasarian–Fromovitz CQ (MFCQ) . . . . . . . 131
5.7.2 Slater CQ . . . . . . . . . . . . . . . . . . . . . . . 131
5.7.3 Linear independence CQ (LICQ) . . . . . . . . . . 132
5.7.4 Affine constraints . . . . . . . . . . . . . . . . . . . 132
5.8 Sufficiency of the KKT conditions under convexity . . . . 133
5.9 Applications and examples . . . . . . . . . . . . . . . . . . 135
5.10 Notes and further reading . . . . . . . . . . . . . . . . . . 137
5.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6 Lagrangian duality 141
6.1 The relaxation theorem . . . . . . . . . . . . . . . . . . . 141
6.2 Lagrangian duality . . . . . . . . . . . . . . . . . . . . . . 142
6.2.1 Lagrangian relaxation and the dual problem . . . . 142
6.2.2 Global optimality conditions . . . . . . . . . . . . 147
6.2.3 Strong duality for convex programs . . . . . . . . . 149
6.2.4 Strong duality for linear and quadratic programs . 154
6.2.5 Two illustrative examples . . . . . . . . . . . . . . 156
6.3 Differentiability properties of the dual function . . . . . . 158
6.3.1 Subdifferentiability of convex functions . . . . . . . 158
6.3.2 Differentiability of the Lagrangian dual function . 162
6.4 ∗Subgradient optimization methods . . . . . . . . . . . . . 164
6.4.1 Convex problems . . . . . . . . . . . . . . . . . . . 164
6.4.2 Application to the Lagrangian dual problem . . . . 170
6.4.3 The generation of ascent directions . . . . . . . . . 173
6.5 ∗Obtaining a primal solution . . . . . . . . . . . . . . . . 174
6.5.1 Differentiability at the optimal solution . . . . . . 175
6.5.2 Everett’s Theorem . . . . . . . . . . . . . . . . . . 176
6.6 ∗Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . 177
6.6.1 Analysis for convex problems . . . . . . . . . . . . 177
6.6.2 Analysis for differentiable problems . . . . . . . . . 179
6.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.7.1 Electrical networks . . . . . . . . . . . . . . . . . . 181
6.7.2 A Lagrangian relaxation of the traveling salesman
problem . . . . . . . . . . . . . . . . . . . . . . . . 185
6.8 Notes and further reading . . . . . . . . . . . . . . . . . . 189
6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
IV Linear Programming 195
7 Linear programming: An introduction 197
7.1 The manufacturing problem . . . . . . . . . . . . . . . . . 197
7.2 A linear programming model . . . . . . . . . . . . . . . . 198
7.3 Graphical solution . . . . . . . . . . . . . . . . . . . . . . 199
7.4 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . 199
7.4.1 An increase in the number of large pieces available 200
7.4.2 An increase in the number of small pieces available 201
7.4.3 A decrease in the price of the tables . . . . . . . . 202
7.5 The dual of the manufacturing problem . . . . . . . . . . 203
7.5.1 A competitor . . . . . . . . . . . . . . . . . . . . . 203
7.5.2 A dual problem . . . . . . . . . . . . . . . . . . . . 203
7.5.3 Interpretations of the dual optimal solution . . . . 204