Ebook: Mathematical Theory of Optimization

Optimization is of central importance in all sciences. Nature inherently seeks optimal solutions. For example, light travels through the "shortest" path and the folded state of a protein corresponds to the structure with the "minimum" potential energy. In combinatorial optimization, there are numerous computationally hard problems arising in real world applications, such as floorplanning in VLSI designs and Steiner trees in communication networks. For these problems, the exact optimal solution is not currently real-time computable. One usually computes an approximate solution with various kinds of heuristics. Recently, many approaches have been developed that link the discrete space of combinatorial optimization to the continuous space of nonlinear optimization through geometric, analytic, and algebraic techniques. Many researchers have found that such approaches lead to very fast and efficient heuristics for solving large problems. Although almost all such heuristics work well in practice there is no solid theoretical analysis, except Karmakar's algorithm for linear programming. With this situation in mind, we decided to teach a seminar on nonlinear optimization with emphasis on its mathematical foundations. This book is the result of that seminar. During the last decades many textbooks and monographs in nonlinear optimization have been published. Why should we write this new one? What is the difference of this book from the others? The motivation for writing this book originated from our efforts to select a textbook for a graduate seminar with focus on the mathematical foundations of optimization.

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This book provides an introduction to the mathematical theory of optimization. It emphasizes the convergence theory of nonlinear optimization algorithms and applications of nonlinear optimization to combinatorial optimization. It includes recent developments in global convergence, the Powell conjecture, semidefinite programming, and relaxation techniques for designs of approximation solutions of combinatorial optimization problems.
Audience: The book can be a textbook or useful reference for undergraduate and graduate students in applied mathematics, operations research, and computer science.

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This book provides an introduction to the mathematical theory of optimization. It emphasizes the convergence theory of nonlinear optimization algorithms and applications of nonlinear optimization to combinatorial optimization. It includes recent developments in global convergence, the Powell conjecture, semidefinite programming, and relaxation techniques for designs of approximation solutions of combinatorial optimization problems.
Audience: The book can be a textbook or useful reference for undergraduate and graduate students in applied mathematics, operations research, and computer science.
Content:
Front Matter....Pages i-xiii
Optimization Problems....Pages 1-21
Linear Programming....Pages 23-40
Blind Man’s Method....Pages 41-50
Hitting Walls....Pages 51-63
Slope and Path Length....Pages 65-79
Average Slope....Pages 81-98
Inexact Active Constraints....Pages 99-123
Efficiency....Pages 125-132
Variable Metric Methods....Pages 133-150
Powell’s Conjecture....Pages 151-166
Minimax....Pages 167-185
Relaxation....Pages 187-200
Semidefinite Programming....Pages 201-213
Interior Point Methods....Pages 215-226
From Local to Global....Pages 227-243
Back Matter....Pages 245-273

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This book provides an introduction to the mathematical theory of optimization. It emphasizes the convergence theory of nonlinear optimization algorithms and applications of nonlinear optimization to combinatorial optimization. It includes recent developments in global convergence, the Powell conjecture, semidefinite programming, and relaxation techniques for designs of approximation solutions of combinatorial optimization problems.
Audience: The book can be a textbook or useful reference for undergraduate and graduate students in applied mathematics, operations research, and computer science.
Content:
Front Matter....Pages i-xiii
Optimization Problems....Pages 1-21
Linear Programming....Pages 23-40
Blind Man’s Method....Pages 41-50
Hitting Walls....Pages 51-63
Slope and Path Length....Pages 65-79
Average Slope....Pages 81-98
Inexact Active Constraints....Pages 99-123
Efficiency....Pages 125-132
Variable Metric Methods....Pages 133-150
Powell’s Conjecture....Pages 151-166
Minimax....Pages 167-185
Relaxation....Pages 187-200
Semidefinite Programming....Pages 201-213
Interior Point Methods....Pages 215-226
From Local to Global....Pages 227-243
Back Matter....Pages 245-273
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