Ioan-Bot Conjugate Duality in Convex Optimization

2010
ISBN: 978-3-642-04900-2
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 637, 164 Seiten

Reihe: Lecture Notes in Economics and Mathematical Systems

The results presented in this book originate from the last decade research work of the author in the ?eld of duality theory in convex optimization. The reputation of duality in the optimization theory comes mainly from the major role that it plays in formulating necessary and suf?cient optimality conditions and, consequently, in generatingdifferent algorithmic approachesfor solving mathematical programming problems. The investigations made in this work prove the importance of the duality theory beyond these aspects and emphasize its strong connections with different topics in convex analysis, nonlinear analysis, functional analysis and in the theory of monotone operators. The ?rst part of the book brings to the attention of the reader the perturbation approach as a fundamental tool for developing the so-called conjugate duality t- ory. The classical Lagrange and Fenchel duality approaches are particular instances of this general concept. More than that, the generalized interior point regularity conditions stated in the past for the two mentioned situations turn out to be p- ticularizations of the ones given in this general setting. In our investigations, the perturbationapproachrepresentsthestartingpointforderivingnewdualityconcepts for several classes of convex optimization problems. Moreover, via this approach, generalized Moreau-Rockafellar formulae are provided and, in connection with them, a new class of regularity conditions, called closedness-type conditions, for both stable strong duality and strong duality is introduced. By stable strong duality we understand the situation in which strong duality still holds whenever perturbing the objective function of the primal problem with a linear continuous functional.

Autoren/Hrsg.

Bot, Radu Ioan

Inhaltsverzeichnis

1;Introduction;12
2;I Perturbation Functions and Dual Problems;20
2.1;1 A General Approach for Duality;20
2.2;2 The Problem Having the Composition with a Linear Continuous Operator in the Objective Function;25
2.3;3 The Problem with Geometric and Cone Constraints;30
2.4;4 The Composed Convex Optimization Problem;39
3;II Moreau–Rockafellar Formulae and Closedness-Type Regularity Conditions;45
3.1;5 Generalized Moreau–Rockafellar Formulae;45
3.2;6 Stable Strong Duality for the Composed ConvexOptimization Problem;48
3.3;7 Stable Strong Duality for the Problem Having the Composition with a Linear Continuous Operator in the Objective Function;54
3.4;8 Stable Strong Duality for the Problem with Geometric and Cone Constraints;60
3.5;9 Closedness Regarding a Set;66
4;III Biconjugate Functions;75
4.1;10 The Biconjugate of a General Perturbation Function;75
4.2;11 Biconjugates Formulae for Different Classesof Convex Functions;78
4.3;12 The Supremum of an (Infinite) Family of ConvexFunctions;83
4.4;13 The Supremum of Two Convex Functions;92
5;IV Strong and Total Conjugate Duality;97
5.1;14 A General Closedness–Type Regularity Condition for (Only) Strong Duality;97
5.2;15 Strong Fenchel Duality;99
5.3;16 Strong Lagrange and Fenchel–Lagrange Duality;103
5.4;17 Total Lagrange and Fenchel–Lagrange Duality;109
6;V Unconventional Fenchel Duality;114
6.1;18 Totally Fenchel Unstable Functions;114
6.2;19 Totally Fenchel Unstable Functions in FiniteDimensional Spaces;121
6.3;20 Quasi Interior and Quasi-relative Interior;124
6.4;21 Regularity Conditions via `39`42`"613A``45`47`"603Aqi and `39`42`"613A``45`47`"603Aqri;128
6.5;22 Lagrange Duality via Fenchel Duality;136
7;VI Applications of the Duality to Monotone Operators;141
7.1;23 Monotone Operators and Their Representative Functions;141
7.2;24 Maximal Monotonicity of the Operator S+A*TA;144
7.3;25 The Maximality of A*TA and S+T;150
7.4;26 Enlargements of Monotone Operators;156
8;References;164
9;Index;170