This book presents new achievements and results in the theory of conjugate duality for convex optimization problems. The perturbation approach for attaching a dual problem to a primal one makes the object of a preliminary chapter, where also an overview of the classical generalized interior point regularity conditions is given. A central role in the book is played by the formulation of generalized Moreau-Rockafellar formulae and closedness-type conditions, the latter constituting a new class of regularity conditions, in many situations with a wider applicability than the generalized interior point ones. The reader also receives deep insights into biconjugate calculus for convex functions, the relations between different existing strong duality notions, but also into several unconventional Fenchel duality topics. The final part of the book is consecrated to the applications of the convex duality theory in the field of monotone operators.
Author(s): Radu Ioan Bot
Series: lecture notes in engeneering economics and Mathematical systems 0637
Edition: 1st Edition.
Publisher: Springer
Year: 2010
Language: English
Pages: 171
cover......Page 1
C.pdf......Page 2
Conjugate Duality in Convex Optimization......Page 3
Introduction......Page 12
1 A General Approach for Duality......Page 20
2 The Problem Having the Composition with a Linear Continuous Operator in the Objective Function......Page 25
3 The Problem with Geometric and Cone Constraints......Page 30
4 The Composed Convex Optimization Problem......Page 39
5 Generalized Moreau–Rockafellar Formulae......Page 45
6 Stable Strong Duality for the Composed ConvexOptimization Problem......Page 48
7 Stable Strong Duality for the Problem Having the Composition with a Linear Continuous Operator in the Objective Function......Page 54
8 Stable Strong Duality for the Problem with Geometric and Cone Constraints......Page 60
9 Closedness Regarding a Set......Page 66
10 The Biconjugate of a General Perturbation Function......Page 75
11 Biconjugates Formulae for Different Classesof Convex Functions......Page 78
12 The Supremum of an (Infinite) Family of ConvexFunctions......Page 83
13 The Supremum of Two Convex Functions......Page 92
14 A General Closedness–Type Regularity Condition for (Only) Strong Duality......Page 97
15 Strong Fenchel Duality......Page 99
16 Strong Lagrange and Fenchel–Lagrange Duality......Page 103
17 Total Lagrange and Fenchel–Lagrange Duality......Page 109
18 Totally Fenchel Unstable Functions......Page 114
19 Totally Fenchel Unstable Functions in FiniteDimensional Spaces......Page 121
20 Quasi Interior and Quasi-relative Interior......Page 124
21 Regularity Conditions via `39`42`"613A``45`47`"603Aqi and `39`42`"613A``45`47`"603Aqri......Page 128
22 Lagrange Duality via Fenchel Duality......Page 136
23 Monotone Operators and Their Representative Functions......Page 141
24 Maximal Monotonicity of the Operator S+A*TA......Page 144
25 The Maximality of A*TA and S+T......Page 150
26 Enlargements of Monotone Operators......Page 156
References......Page 164
Index......Page 170
