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E-raamat: Set-valued Optimization: An Introduction with Applications

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  • Formaat: PDF+DRM
  • Sari: Vector Optimization
  • Ilmumisaeg: 20-Oct-2014
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783642542657
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Set-valued Optimization: An Introduction with ApplicationsSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Vector Optimization
  • Ilmumisaeg: 20-Oct-2014
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783642542657
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Set-valued optimization is a vibrant and expanding branch of mathematics that deals with optimization problems where the objective map and/or the constraints maps are set-valued maps acting between certain spaces. Since set-valued maps subsumes single valued maps, set-valued optimization provides an important extension and unification of the scalar as well as the vector optimization problems. Therefore this relatively new discipline has justifiably attracted a great deal of attention in recent years. This book presents, in a unified framework, basic properties on ordering relations, solution concepts for set-valued optimization problems, a detailed description of convex set-valued maps, most recent developments in separation theorems, scalarization techniques, variational principles, tangent cones of first and higher order, sub-differential of set-valued maps, generalized derivatives of set-valued maps, sensitivity analysis, optimality conditions, duality and applications in economics among other things.

Arvustused

In this monograph, the first one entirely devoted to the subject, the authors present a unified treatment of the state of the art of most of the developments in this field. It is very well written and contains numerous results and examples and over 600 references. I wholeheartedly recommend this book to anyone interested in new trends in optimization, in particular, researchers who are interested in multiobjective/vector-optimization and its interconnection with variational analysis. (Miguel Sama, Mathematical Reviews, December, 2015)

The monograph contains a very large number of interesting, new or at least recent results not only in the fields of set-valued optimization together with many examples illustrating them. It is an interesting scientific resource for all researchers investigating problems of mathematical optimization.I can highly recommend it to graduate and PhD students as well as to scientists working in the fields of nonsmooth, multicriterial or set-valued optimization. (S. Dempe, Optimization, July, 2015)

This massive and very well-written book will probably be for many years the primary reference in set-valued optimization, the area of optimization where the objective and/or constraint functions of the considered problems are set-valued maps. this book will be a solid reference for everyone interested in set-valued optimization, be they graduate students or established scientists. (Sorin-Mihai Grad, zbMATH 1308.49004, 2015)

1 Introduction 1(10)
1.1 Motivating Examples
1(3)
1.2 Book Structure
4(5)
1.3 Useful Notation
9(2)
2 Order Relations and Ordering Cones 11(66)
2.1 Order Relations
11(6)
2.2 Cone Properties Related to the Topology and the Order
17(5)
2.3 Convexity Notions for Sets and Set-Valued Maps
22(6)
2.4 Solution Concepts in Vector Optimization
28(15)
2.5 Vector Optimization Problems with Variable Ordering Structure
43(2)
2.6 Solution Concepts in Set-Valued Optimization
45(29)
2.6.1 Solution Concepts Based on Vector Approach
45(3)
2.6.2 Solution Concepts Based on Set Approach
48(7)
2.6.3 Solution Concepts Based on Lattice Structure
55(10)
2.6.4 The Embedding Approach by Kuroiwa
65(2)
2.6.5 Solution Concepts with Respect to Abstract Preference Relations
67(3)
2.6.6 Set-Valued Optimization Problems with Variable Ordering Structure
70(3)
2.6.7 Approximate Solutions of Set-Valued Optimization Problems
73(1)
2.7 Relationships Between Solution Concepts
74(3)
3 Continuity and Differentiability 77(32)
3.1 Continuity Notions for Set-Valued Maps
77(13)
3.2 Continuity Properties of Set-Valued Maps Under Convexity Assumptions
90(6)
3.3 Lipschitz Properties for Single-Valued and Set-Valued Maps
96(6)
3.4 Clarke's Normal Cone and Subdifferential
102(1)
3.5 Limiting Cones and Generalized Differentiability
103(4)
3.6 Approximate Cones and Generalized Differentiability
107(2)
4 Tangent Cones and Tangent Sets 109(104)
4.1 First-Order Tangent Cones
110(13)
4.1.1 The Radial Tangent Cone and the Feasible Tangent Cone
110(2)
4.1.2 The Contingent Cone and the Interiorly Contingent Cone
112(8)
4.1.3 The Adjacent Cone and the Interiorly Adjacent Cone
120(3)
4.2 Modified First-Order Tangent Cones
123(6)
4.2.1 The Modified Radial and the Modified Feasible Tangent Cones
124(1)
4.2.2 The Modified Contingent and the Modified Interiorly Contingent Cones
124(2)
4.2.3 The Modified Adjacent and the Modified Interiorly Adjacent Cones
126(3)
4.3 Miscellaneous Properties of First-Order Tangent Cones
129(3)
4.4 First-Order Tangent Cones on Convex Sets
132(11)
4.4.1 Connections Among First-Order Tangent Cones on Convex Sets
132(5)
4.4.2 Properties of First-Order Tangent Cones on Convex Sets
137(6)
4.5 First-Order Local Cone Approximation
143(4)
4.6 Convex Subcones of the Contingent Cone
147(9)
4.7 First-Order Inversion Theorems and Intersection Formulas
156(5)
4.8 Expressions of the Contingent Cone on Some Constraint Sets
161(8)
4.9 Second-Order Tangent Sets
169(6)
4.9.1 Second-Order Radial Tangent Set and Second-Order Feasible Tangent Set
170(1)
4.9.2 Second-Order Contingent Set and Second-Order Interiorly Contingent Set
170(3)
4.9.3 Second-Order Adjacent Set and Second-Order Interiorly Adjacent Set
173(2)
4.10 Generalized Second-Order Tangent Sets
175(6)
4.11 Second-Order Asymptotic Tangent Cones
181(6)
4.11.1 Second-Order Asymptotic Feasible Tangent Cone and Second-Order Asymptotic Radial Tangent Cone
182(1)
4.11.2 Second-Order Asymptotic Contingent Cone and Second-Order Asymptotic Interiorly Contingent Cone
183(2)
4.11.3 Second-Order Asymptotic Adjacent Cone and Second-Order Asymptotic Interiorly Adjacent Cone
185(2)
4.12 Miscellaneous Properties of Second-Order Tangent Sets and Second-Order Asymptotic Tangent Cones
187(5)
4.13 Second-Order Inversion Theorems
192(5)
4.14 Expressions of the Second-Order Contingent Set on Specific Constraints
197(5)
4.15 Miscellaneous Second-Order Tangent Cones
202(5)
4.15.1 Second-Order Tangent Cones of Ledzewicz and Schaettler
202(2)
4.15.2 Projective Tangent Cones of Second-Order
204(2)
4.15.3 Second-Order Tangent Cone of N. Pavel
206(1)
4.15.4 Connections Among the Second-Order Tangent Cones
207(1)
4.16 Second-Order Local Approximation
207(3)
4.17 Higher-Order Tangent Cones and Tangent Sets
210(3)
5 Nonconvex Separation Theorems 213(36)
5.1 Separating Functions and Examples
213(4)
5.2 Nonlinear Separation
217(15)
5.2.1 Construction of Scalarizing Functionals
217(2)
5.2.2 Properties of Scalarization Functions
219(5)
5.2.3 Continuity Properties
224(1)
5.2.4 Lipschitz Properties
225(6)
5.2.5 The Formula for the Conjugate and Subdifferential of φA for A Convex
231(1)
5.3 Scalarizing Functionals by Hiriart-Urruty and Zaffaroni
232(4)
5.4 Characterization of Solutions of Set-Valued Optimization Problems by Means of Nonlinear Scalarizing Functionals
236(8)
5.4.1 An Extension of the Functional cpA
236(4)
5.4.2 Characterization of Solutions of Set-Valued Optimization Problems with Lower Set Less Order Relation > or = to C by Scalarization
240(4)
5.5 The Extremal Principle
244(5)
6 Hahn-Banach Type Theorems 249(26)
6.1 The Hahn-Banach-Kantorovich Theorem
250(8)
6.2 Classical Separation Theorems for Convex Sets
258(3)
6.3 The Core Convex Topology
261(3)
6.4 Yang's Generalization of the Hahn-Banach Theorem
264(7)
6.5 A Sufficient Condition for the Convexity of R+A
271(4)
7 Conjugates and Subdifferentials 275(32)
7.1 The Strong Conjugate and Subdifferential
275(13)
7.2 The Weak Subdifferential
288(8)
7.3 Subdifferentials Corresponding to Henig Proper Efficiency
296(2)
7.4 Exact Formulas for the Subdifferential of the Sum and the Composition
298(9)
8 Duality 307(42)
8.1 Duality Assertions for Set-Valued Problems Based on Vector Approach
308(9)
8.1.1 Conjugate Duality for Set-Valued Problems Based on Vector Approach
308(5)
8.1.2 Lagrange Duality for Set-Valued Optimization Problems Based on Vector Approach
313(4)
8.2 Duality Assertions for Set-Valued Problems Based on Set Approach
317(5)
8.3 Duality Assertions for Set-Valued Problems Based on Lattice Structure
322(16)
8.3.1 Conjugate Duality for F-Valued Problems
323(3)
8.3.2 Lagrange Duality for F-Valued Problems
326(12)
8.4 Comparison of Different Approaches to Duality in Set-Valued Optimization
338(11)
8.4.1 Lagrange Duality
339(2)
8.4.2 Subdifferentials and Stability
341(4)
8.4.3 Duality Statements with Operators as Dual Variables
345(4)
9 Existence Results for Minimal Points 349(20)
9.1 Preliminary Notions and Results Concerning Transitive Relations
349(3)
9.2 Existence of Minimal Elements with Respect to Transitive Relations
352(3)
9.3 Existence of Minimal Points with Respect to Cones
355(5)
9.4 Types of Convex Cones and Compactness with Respect to Cones
360(2)
9.5 Existence of Optimal Solutions for Vector and Set Optimization Problems
362(7)
10 Ekeland Variational Principle 369(30)
10.1 Preliminary Notions and Results
369(4)
10.2 Minimal Points in Product Spaces
373(8)
10.3 Minimal Points in Product Spaces of Isac-Tammer's Type
381(3)
10.4 Ekeland's Variational Principles of Ha's Type
384(6)
10.5 Ekeland's Variational Principle for Bi-Set-Valued Maps
390(1)
10.6 EVP Type Results
391(3)
10.7 Error Bounds
394(5)
11 Derivatives and Epiderivatives of Set-Valued Maps 399(110)
11.1 Contingent Derivatives of Set-Valued Maps
400(28)
11.1.1 Miscellaneous Graphical Derivatives of Set-valued Maps
407(7)
11.1.2 Convexity Characterization Using Contingent Derivatives
414(2)
11.1.3 Proto-Differentiability, Semi-Differentiability, and Related Concepts
416(6)
11.1.4 Weak Contingent Derivatives of Set-Valued Maps
422(4)
11.1.5 A Lyusternik-Type Theorem Using Contingent Derivatives
426(2)
11.2 Calculus Rules for Derivatives of Set-Valued Maps
428(9)
11.2.1 Calculus Rules by a Direct Approach
429(3)
11.2.2 Derivative Rules by Using Calculus of Tangent Cones
432(5)
11.3 Contingently C -Absorbing Maps
437(8)
11.4 Epiderivatives of Set-Valued Maps
445(25)
11.4.1 Contingent Epiderivatives of Set-Valued Maps with Images in R
446(6)
11.4.2 Contingent Epiderivatives in General Spaces
452(5)
11.4.3 Existence Theorems for Contingent Epiderivatives
457(7)
11.4.4 Variational Characterization of the Contingent Epiderivatives
464(6)
11.5 Generalized Contingent Epiderivatives of Set-Valued Maps
470(12)
11.5.1 Existence Theorems for Generalized Contingent Epiderivatives
474(4)
11.5.2 Characterizations of Generalized Contingent Epiderivatives
478(4)
11.6 Calculus Rules for Contingent Epiderivatives
482(6)
11.7 Second-Order Derivatives of Set-Valued Maps
488(12)
11.8 Calculus Rules for Second-Order Contingent Derivatives
500(4)
11.9 Second-Order Epiderivatives of Set-Valued Maps
504(5)
12 Optimality Conditions in Set-Valued Optimization 509(96)
12.1 First-Order Optimality Conditions by the Direct Approach
512(10)
12.2 First-Order Optimality Conditions by the Dubovitskii-Milyutin Approach
522(20)
12.2.1 Necessary Optimality Conditions by the Dubovitskii-Milyutin Approach
523(4)
12.2.2 Inverse Images and Subgradients of Set-Valued Maps
527(7)
12.2.3 Separation Theorems and the Dubovitskii-Milyutin Lemma
534(3)
12.2.4 Lagrange Multiplier Rules by the Dubovitskii-Milyutin Approach
537(5)
12.3 Sufficient Optimality Conditions in Set-Valued Optimization
542(7)
12.3.1 Sufficient Optimality Conditions Under Convexity and Quasi-Convexity
542(3)
12.3.2 Sufficient Optimality Conditions Under Paraconvexity
545(4)
12.3.3 Sufficient Optimality Conditions Under Semidifferentiability
549(1)
12.4 Second-Order Optimality Conditions in Set-Valued Optimization
549(8)
12.4.1 Second-Order Optimality Conditions by the Dubovitskii-Milyutin Approach
550(4)
12.4.2 Second-Order Optimality Conditions by the Direct Approach
554(3)
12.5 Generalized Dubovitskii-Milyutin Approach in Set-Valued Optimization
557(11)
12.5.1 A Separation Theorem for Multiple Closed and Open Cones
559(3)
12.5.2 First-Order Generalized Dubovitskii-Milyutin Approach
562(5)
12.5.3 Second-Order Generalized Dubovitskii-Milyutin Approach
567(1)
12.6 Set-Valued Optimization Problems with a Variable Order Structure
568(4)
12.7 Optimality Conditions for Q-Minimizers in Set-Valued Optimization
572(6)
12.7.1 Optimality Conditions for Q-Minimizers Using Radial Derivatives
572(2)
12.7.2 Optimality Conditions for Q-Minimizers Using Coderivatives
574(4)
12.8 Lagrange Multiplier Rules Based on Limiting Subdifferential
578(13)
12.9 Necessary Conditions for Approximate Solutions of Set-Valued Optimization Problems
591(3)
12.10 Necessary and Sufficient Conditions for Solution Concepts Based on Set Approach
594(4)
12.11 Necessary Conditions for Solution Concepts with Respect to a General Preference Relation
598(2)
12.12 KKT-Points and Corresponding Stability Results
600(5)
13 Sensitivity Analysis in Set-Valued Optimization and Vector Variational Inequalities 605(40)
13.1 First Order Sensitivity Analysis in Set-Valued Optimization
606(7)
13.2 Second Order Sensitivity Analysis in Set-Valued Optimization
613(10)
13.3 Sensitivity Analysis in Set-Valued Optimization Using Coderivatives
623(11)
13.4 Sensitivity Analysis for Vector Variational Inequalities
634(11)
14 Numerical Methods for Solving Set-Valued Optimization Problems 645(18)
14.1 A Newton Method for Set-Valued Maps
645(6)
14.2 An Algorithm to Solve Polyhedral Convex Set-Valued Optimization Problems
651(12)
14.2.1 Formulation of the Polyhedral Convex Set-Valued Optimization Problem
653(2)
14.2.2 An Algorithm for Solving Polyhedral Convex Set-Valued Optimization Problems
655(3)
14.2.3 Properties of the Algorithm
658(5)
15 Applications 663(64)
15.1 Set-Valued Approaches to Duality in Vector Optimization
663(33)
15.1.1 Fenchel Duality for Vector Optimization Problems Using Corresponding Results for F-Valued Problems
667(3)
15.1.2 Lagrange Duality for Vector Optimization Problems Based on Results for J-Valued Problems
670(7)
15.1.3 Duality Assertions for Linear Vector Optimization Based on Lattice Approach
677(5)
15.1.4 Further Set-Valued Approaches to Duality in Linear Vector Optimization
682(14)
15.2 Applications in Mathematical Finance
696(5)
15.3 Set-Valued Optimization in Welfare Economics
701(5)
15.4 Robustness for Vector-Valued Optimization Problems
706(21)
15.4.1 > or = to uC - Robustness
710(10)
15.4.2 > or = to lC - Robustness
720(2)
15.4.3 > or = to sC - Robustness
722(2)
15.4.4 Algorithms for Solving Special Classes of Set-Valued Optimization Problems
724(3)
Appendix 727(6)
References 733(26)
Index 759
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