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Convexity and Optimization in Rn



Convexity and Optimization in Rn

Leonard D. Berkovitz

ISBN: 978-0-471-46166-1 April 2003 280 Pages

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A comprehensive introduction to convexity and optimization inRn

This book presents the mathematics of finite dimensionalconstrained optimization problems. It provides a basis for thefurther mathematical study of convexity, of more generaloptimization problems, and of numerical algorithms for the solutionof finite dimensional optimization problems. For readers who do nothave the requisite background in real analysis, the author providesa chapter covering this material. The text features abundantexercises and problems designed to lead the reader to a fundamentalunderstanding of the material.

Convexity and Optimization in Rn provides detailed discussionof:
* 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 indexoffers quick reference. Suitable as a text for both graduate andundergraduate students in mathematics and engineering, thisaccessible text is written from extensively class-tested notes.

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 NonlinearProgrammingProblems.

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.


"...a nice introduction to finite-dimensional optimization..."(Zentralblatt Math, Vol.991, No.16, 2002)

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

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