Convexity and Optimization in Rn

284 Pages·2001·1.485 MB·English

by  Leonard D. Berkovitz

CONVEXITY AND
R(cid:1)
OPTIMIZATION IN
LEONARDD. BERKOVITZ
Purdue University
A Wiley-IntersciencePublication
JOHN WILEY & SONS, INC.

CONVEXITY AND
R(cid:1)
OPTIMIZATION IN

CONVEXITY AND
R(cid:1)
OPTIMIZATION IN
LEONARDD. BERKOVITZ
Purdue University
A Wiley-IntersciencePublication
JOHN WILEY & SONS, INC.
(cid:1)
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LibraryofCongressCataloging-in-PublicationData:
Berkovitz,LeonardDavid,1924—
ConvexityandoptimizationinR[superscriptn]/LeonardD.Berkovitz.
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‘‘AWiley-Intersciencepublication.’’
Includesbibliographicalreferencesandindex.
ISBN0-471-35281-0(cloth:alk.paper)
1.Convexsets.2.Mathematicaloptimization.I.Title.II.Pureandapplied
mathematics(JohnWiley&Sons:Unnumbered)
QA640.B462001
516(cid:1).08--dc21
2001045391
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To my wife, Anna

CONTENTS
Preface ix
I Topics in Real Analysis 1
1. Introduction 1
2. Vectors in R(cid:1) 1
3. Algebra of Sets 4
4. Metric Topology of R(cid:1) 5
5. Limits and Continuity 8
6. Basic Property of Real Numbers 11
7. Compactness 14
8. Equivalent Norms and Cartesian Products 16
9. Fundamental Existence Theorem 18
10. Linear Transformations 21
11. Differentiation in R(cid:1) 22
II Convex Sets in RRn 30
1. Lines and Hyperplanes in R(cid:1) 30
2. Properties of Convex Sets 35
3. Separation Theorems 45
4. Supporting Hyperplanes: Extreme Points 56
5. Systems of Linear Inequalities: Theorems of the Alternative 61
6. Affine Geometry 69
7. More on Separation and Support 80
III Convex Functions 87
1. Definition and Elementary Properties 87
2. Subgradients 102
3. Differentiable Convex Functions 106
4. Alternative Theorems for Convex Functions 113
5. Application to Game Theory 117
vii