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Convexity and Optimization in Rn

ISBN: 978-0-471-35281-5
280 pages
December 2001
Convexity and Optimization in Rn (0471352810) cover image
A comprehensive introduction to convexity and optimization in Rn

This book presents the mathematics of finite dimensional constrained optimization problems. It provides a basis for the further mathematical study of convexity, of more general optimization problems, and of numerical algorithms for the solution of finite dimensional optimization problems. For readers who do not have the requisite background in real analysis, the author provides a chapter covering this material. The text features abundant exercises and problems designed to lead the reader to a fundamental understanding of the material.

Convexity and Optimization in Rn provides detailed discussion of:
* Requisite topics in real analysis
* Convex sets
* Convex functions
* Optimization problems
* Convex programming and duality
* The simplex method


A detailed bibliography is included for further study and an index offers quick reference. Suitable as a text for both graduate and undergraduate students in mathematics and engineering, this accessible text is written from extensively class-tested notes.
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Preface.

I: Topics in Real Analysis.

1. Introduction.

2. Vectors in R".

3. Algebra of Sets.

4. Metric Topology of R".

5. Limits and Continuity.

6. Basic Propertyof Real Numbers.

7. Compactness.

8. Equivalent Norms and Cartesian Products.

9. Fundamental Existence Theorem.

10. Linear Transformations.

11. Differentiation in R".

II: Convex Sets in R".

1. Lines and Hyperplanes in R".

2. Properties of Convex Sets.

3. Separation Theorems.

4. Supporting Hyperplanes:Extreme Points.

5. Systems of Linear Inequalities:Theorems of the Alternative .

6. Affine Geometry.

7. More on Separation and Support.

III: Convex Functions.

1. Definition and Elementary Properties.

2. Subgradients.

3. Differentiable Convex Functions.

4. Alternative Theorems for Convex Functions.

5. Application to Game Theory.

IV: Optimization Problems.

1. Introduction.

2. Differentiable Unconstrained Problems.

3. Optimization of Convex Functions.

4. Linear Programming Problems.

5. First-Order Conditions for Differentiable NonlinearProgramming Problems.

6. Second-Order Conditions.

V: Convex Programming and Duality.

1. Problem Statement.

2. Necessary Conditions and Sufficient Conditions.

3. Perturbation Theory.

4. Lagrangian Duality.

5. Geometric Interpretation.

6. Quadratic Programming.

7. Dualityin Linear Programming.

VI: Simplex Method.

1. Introduction.

2. Extreme Points of Feasible Set.

3. Preliminaries to Simplex Method.

4. Phase II of Simplex Method.

5. Termination and Cycling.

6. Phase I of Simplex Method.

7. Revised Simplex Method.

Bibliography.

Index.
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LEONARD D. BERKOVITZ, PhD, is Professor of Mathematics at Purdue University. He previously worked at the RAND Corporation and has served on the editorial boards of several journals, including terms as Managing Editor of the SIAM Journal on Control and as a member of the Editorial Committee of Mathematical Reviews.
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"...a nice introduction to finite-dimensional optimization..." (Zentralblatt Math, Vol.991, No.16, 2002)

"A textbook for a one-semester...course for students of engineering, economics, operations research, and mathematics." (SciTech Book News, Vol. 26, No. 2, June 2002)

"...a fine introductory textbook that provides a solid introduction to the subject as well as a good foundation for further study..." (Mathematical Reviews, 2003a)
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