A Gentle Introduction to Optimization 1st Edition by Guenin, Könemann, Tunçel ISBN 978-1107658790 1107658799

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Product details: 

ISBN 10:  1107658799

ISBN 13:  978-1107658790

Author:  B. Guenin,  J. Könemann, L. Tunçel

Optimization is an essential technique for solving problems in areas as diverse as accounting, computer science and engineering. Assuming only basic linear algebra and with a clear focus on the fundamental concepts, this textbook is the perfect starting point for first- and second-year undergraduate students from a wide range of backgrounds and with varying levels of ability. Modern, real-world examples motivate the theory throughout. The authors keep the text as concise and focused as possible, with more advanced material treated separately or in starred exercises. Chapters are self-contained so that instructors and students can adapt the material to suit their own needs and a wide selection of over 140 exercises gives readers the opportunity to try out the skills they gain in each section. Solutions are available for instructors. The book also provides suggestions for further reading to help students take the next step to more advanced material.

Table of contents: 

1 Introduction

1.1 A first example

1.1.1 The formulation

1.1.2 Correctness

1.2 Linear programs

1.2.1 Multiperiod models

1.3 Integer programs

1.3.1 Assignment problem

1.3.2 Knapsack problem

1.4 Optimization problems on graphs

1.4.1 Shortest path problem

1.4.2 Minimum cost perfect matching

1.5 Integer programs continued

1.5.1 Minimum cost perfect matching

1.5.2 Shortest path problem

1.6 Nonlinear programs

1.6.1 Pricing a tech gadget

1.6.2 Finding a closest point feasible in an LP

1.6.3 Finding a “central” feasible solution of an LP

1.7 Overview of the book

1.8 Further reading and notes

2 Solving linear programs

2.1 Possible outcomes

2.1.1 Infeasible linear programs

2.1.2 Unbounded linear programs

2.1.3 Linear programs with optimal solutions

2.2 Standard equality form

2.3 A simplex iteration

2.4 Bases and canonical forms

2.4.1 Bases

2.4.2 Canonical forms

2.5 The simplex algorithm

2.5.1 An example with an optimal solution

2.5.2 An unbounded example

2.5.3 Formalizing the procedure

2.6 Finding feasible solutions

2.6.1 General scheme

2.6.2 The two phase simplex algorithm-an example

2.6.3 Consequences

2.7 Simplex via tableaus*

2.7.1 Pivoting

2.7.2 Tableaus

2.8 Geometry

2.8.1 Feasible region of LPs and polyhedra

2.8.2 Convexity

2.8.3 Extreme points

2.8.4 Geometric interpretation of the simplex algorithm

2.9 Further reading and notes

3 Duality through examples

3.1 The shortest path problem

3.1.1 An intuitive lower bound

3.1.2 A general argument – weak duality

3.1.3 Revisiting the intuitive lower bound

3.1.4 An algorithm

3.1.5 Correctness of the algorithm

3.2 Minimum cost perfect matching in bipartite graphs

3.2.1 An intuitive lower bound

3.2.2 A general argument-weak duality

3.2.3 Revisiting the intuitive lower bound

3.2.4 An algorithm

3.2.5 Correctness of the algorithm

3.2.6 Finding perfect matchings in bipartite graphs*

3.3 Further reading and notes

4 Duality theory

4.1 Weak duality

4.2 Strong duality

4.3 A geometric characterization of optimality

4.3.1 Complementary slackness

4.3.2 Geometry

4.4 Wolf’s lemma*

4.5 Further reading and notes

5 Applications of duality*

5.1 Approximation algorithm for set-cover

5.1.1 A primal-dual algorithm

5.1.2 Greed is good at least sometimes

5.1.3 Discussion

5.2 Economic interpretation

5.3 The maximum-flow-minimum-cut theorem

5.3.1 Totally unimodular matrices

5.3.2 Applications to st-flows

6 Solving integer programs

6.1 Integer programs versus linear programs

6.2 Cutting planes

6.2.1 Cutting planes and the simplex algorithm

6.3 Branch and bound

6.4 Traveling salesman problem and a separation algorithm*

6.5 Further reading and notes

7 Nonlinear optimization

7.1 Some examples

7.2 Some nonlinear programs are very hard

7.2.1 NLP versus 0,1 integer programming

7.2.2 Hard small-dimensional instances

7.3 Convexity

7.3.1 Convex functions and epigraphs

7.3.2 Level sets and feasible region

7.4 Relaxing convex NLPs

7.4.1 Subgradients

7.4.2 Supporting halfspaces

7.5 Optimality conditions for the differentiable case

7.5.1 Sufficient conditions for optimality

7.5.2 Differentiability and gradients

7.5.3 A Karush-Kuhn-Tucker theorem

7.6 Optimality conditions based on Lagrangians

7.7 Nonconvex optimization problems

7.7.1 The Karush-Kuhn-Tucker theorem for nonconvex problems

7.7.2 Convex relaxation approach to nonconvex problems*

7.8 Interior-point method for linear programs*

7.8.1 A polynomial-time interior-point algorithm*

7.9 Further reading and notes

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Tags: Guenin, Könemann, Tunçel, A Gentle Introduction