An Optimization Primer: On Models, Algorithms, and Duality

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Springer Science & Business Media, May 18, 2004 - Mathematics - 108 pages
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Optimization is the art, science and mathematics of finding the "best" member of a finite or infinite set of possible choices, based on some objective measure of the merit of each choice in the set. Three key facets of the subject are:

- the construction of optimization models that capture the range of available choices within a feasible set and the measure-of-merit of any particular choice in a feasible set relative to its competitors;

- the invention and implementation of efficient algorithms for solving optimization models;

- a mathematical principle of duality that relates optimization models to one another in a fundamental way. Duality cuts across the entire field of optimization and is useful, in particular, for identifying optimality conditions, i.e., criteria that a given member of a feasible set must satisfy in order to be an optimal solution.

This booklet provides a gentle introduction to the above topics and will be of interest to college students taking an introductory course in optimization, high school students beginning their studies in mathematics and science, the general reader looking for an overall sense of the field of optimization, and specialists in optimization interested in developing new ways of teaching the subject to their students.

John Lawrence Nazareth is Professor Emeritus in the Department of Mathematics at Washington State University and Affiliate Professor in the Department of Applied Mathematics at the University of Washington. He is the author of two recent books also published by Springer-Verlag which explore the above topics in more depth, Differentiable Optimization and Equation Solving (2003) and DLP and Extensions: An Optimization Model and Decision Support System (2001).

 

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Contents

Simple Motivating Examples
1
12 Watering the Garden
2
13 Chopping Wood
3
14 Going Fishing
4
15 Summary
5
A Quintessential Optimization Problem
7
22 Algorithms
8
23 Duality
14
64 Notes
53
An Algorithmic Revolution
55
71 Affine Scaling
56
72 Central Path
60
73 InteriorPoint Algorithms
64
74 Notes
68
Nonlinear Programming
69
81 Geometric Perspective
70

24 Notes
18
Duality on Bipartite Networks
19
31 Matching
21
32 Covering
23
33 KönigEgerváry Duality
27
34 Notes
28
A Network Flow Overview
29
42 A Network Flow Tree
31
Combinatorial visŕvis Continuous
34
44 Notes
35
Duality in Linear Programming
37
52 Farkas Duality and LP Optimality
40
53 Notes
44
The Golden Age of Optimization
45
62 Linear Programming in Practice
50
63 Network Simplex Algorithm
52
82 Algebraic Perspective
72
83 Information Costs
73
84 Dimensions
74
85 Constraints
75
86 Differentiable Programming
76
87 Notes
78
DLP and Extensions
79
92 A Rangeland Improvement Problem
82
93 ResourceDecision Software
86
94 Notes
91
Optimization The Big Picture
93
102 Notes
97
References
99
Index
105
About the Author
109
Copyright

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Page 102 - The analogue of Moreau's proximation theorem, with applications to the nonlinear complementary problem," Pacific Journal of Mathematics, 88, 101-161. [42] Megiddo, N. (1989), "Pathways to the optimal set in linear programming," in Progress in Mathematical Programming: Interior- Point and Related Methods, N.

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