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Finite dimensional convexity and optimization / Monique Florenzano and Cuong Le Van, in cooperation with Pascal Gourdel.

By: Contributor(s): Series: Studies in economic theory ; 13.Publication details: Berlin : Springer-Verlag, 2001.Description: xii, 154 p. : illustrations ; 24 cmISBN:
  • 9783642625701
Subject(s): DDC classification:
  • 515.8 23 F633
Contents:
1. Convexity in R[superscript n] -- 1.1. Basic concepts -- 1.2. Topological properties of convex sets -- 2. Separation and Polarity -- 2.1. Separation of convex sets -- 2.2. Polars of convex sets and orthogonal subspaces -- 2.3. Application to Minkowski-Farkas' lemma -- 3. Extremal Structure of Convex Sets -- 3.1. Extreme points and faces of convex sets -- 3.2. Application to linear inequalities. Weyl's theorem -- 3.3. Extreme points and extremal subsets of a polyhedral convex set -- 4. Linear Programming -- 4.1. Necessary and sufficient conditions of optimality -- 4.2. The duality theorem of linear programming -- 4.3. The simplex method -- 5. Convex Functions -- 5.1. Basic definitions and properties -- 5.2. Continuity theorems -- 5.3. Continuity properties of collections of convex functions -- 6. Differential Theory of Convex Functions -- 6.1. The Hahn-Banach dominated extension theorem -- 6.2. Sublinear functions -- 6.3. Support functions -- 6.4. Directional derivatives -- 6.5. Subgradients and subdifferential of a convex function -- 6.6. Differentiability of convex functions -- 6.7. Differential continuity for convex functions -- 7. Convex Optimization With Convex Constraints -- 7.1. The minimum of a convex function f: R[superscript n] [approaches]R -- 7.2. Kuhn-Tucker Conditions -- 7.3. Value functions -- 8. Non Convex Optimization -- 8.1. Quasi-convex functions -- 8.2. Minimization of quasi-convex functions -- 8.3. Differentiable optimization -- A. Appendix-- Bibliography-- Index.
Summary: The primary aim of this book is to present notions of convex analysis which constitute the basic underlying structure of argumentation in economic theory and which are common to optimization problems encountered in many applications. The intended readers are graduate students, and specialists of mathematical programming whose research fields are applied mathematics and economics. The text consists of a systematic development in eight chapters, with guided exercises containing sometimes significant and useful additional results. The book is appropriate as a class text, or for self-study.
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Holdings
Item type Current library Call number Status Date due Barcode Item holds
Books ISI Library, Kolkata 515.8 F633 (Browse shelf(Opens below)) Available 136379
Total holds: 0

Includes bibliographical references and index.

1. Convexity in R[superscript n] --
1.1. Basic concepts --
1.2. Topological properties of convex sets --
2. Separation and Polarity --
2.1. Separation of convex sets --
2.2. Polars of convex sets and orthogonal subspaces --
2.3. Application to Minkowski-Farkas' lemma --
3. Extremal Structure of Convex Sets --
3.1. Extreme points and faces of convex sets --
3.2. Application to linear inequalities. Weyl's theorem --
3.3. Extreme points and extremal subsets of a polyhedral convex set --
4. Linear Programming --
4.1. Necessary and sufficient conditions of optimality --
4.2. The duality theorem of linear programming --
4.3. The simplex method --
5. Convex Functions --
5.1. Basic definitions and properties --
5.2. Continuity theorems --
5.3. Continuity properties of collections of convex functions --
6. Differential Theory of Convex Functions --
6.1. The Hahn-Banach dominated extension theorem --
6.2. Sublinear functions --
6.3. Support functions --
6.4. Directional derivatives --
6.5. Subgradients and subdifferential of a convex function --
6.6. Differentiability of convex functions --
6.7. Differential continuity for convex functions --
7. Convex Optimization With Convex Constraints --
7.1. The minimum of a convex function f: R[superscript n] [approaches]R --
7.2. Kuhn-Tucker Conditions --
7.3. Value functions --
8. Non Convex Optimization --
8.1. Quasi-convex functions --
8.2. Minimization of quasi-convex functions --
8.3. Differentiable optimization --
A. Appendix--
Bibliography--
Index.

The primary aim of this book is to present notions of convex analysis which constitute the basic underlying structure of argumentation in economic theory and which are common to optimization problems encountered in many applications. The intended readers are graduate students, and specialists of mathematical programming whose research fields are applied mathematics and economics. The text consists of a systematic development in eight chapters, with guided exercises containing sometimes significant and useful additional results. The book is appropriate as a class text, or for self-study.

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