Variational Methods in Partially Ordered Spaces

by Gopfert, Alfred; Tammer, Christiane; Riahi, Hassan; Zalinescu, Constantin; Riahi, Hassan

• ISBN13: 9780387004525
• ISBN10: 0387004521
• Format: Hardcover
• Copyright: 2003-08-01
• Publisher: Springer Nature

Summary

In mathematical modeling of processes one often encountersoptimization problems involving more than one objective function, sothat Multiobjective Optimization (or Vector Optimization) has receivednew impetus. The growing interest in multiobjective problems, bothfrom the theoretical point of view and as it concerns applications toreal problems, asks for a general scheme which embraces severalexisting developments and stimulates new ones. In this book theauthors provide the newest results and applications of this quicklygrowing field.This book will be of interest to graduate students in mathematics,economics, and engineering, as well as researchers in pure and appliedmathematics, economics, engineering, geography, and town planning. Asound knowledge of linear algebra and introductory real analysisshould provide readers with sufficient background for this book.

Table of Contents

Preface	v
List of Figures	xiii
Examples	1	(12)
Cones in Vector Spaces	1	(2)
Equilibrium Problems	3	(3)
Location Problems in Town Planning	6	(2)
Multicriteria Control Problems	8	(2)
Multicriteria Fractional Programming Problems	10	(1)
Stochastic Efficiency in a Set	11	(2)
Functional Analysis over Cones	13	(68)
Order Structures	13	(13)
Functional Analysis and Convexity	26	(13)
Separation Theorems for Not Necessarily Convex Sets	39	(6)
Convexity Notions for Sets and Multifunctions	45	(6)
Continuity Notions for Multifunctions	51	(19)
Continuity Properties of Multifunctions Under Convexity Assumptions	70	(3)
Tangent Cones and Differentiability of Multifunctions	73	(8)
Optimization in Partially Ordered Spaces	81	(148)
Solution Concepts	81	(5)
Approximate Minimality	81	(3)
A General Scalarization Method	84	(2)
Existence Results for Efficient Points	86	(29)
Preliminary Notions and Results Concerning Transitive Relations	87	(2)
Existence of Maximal Elements with Respect to Transitive Relations	89	(5)
Existence of Efficient Points with Respect to Cones	94	(7)
Types of Convex Cones and Compactness with Respect to Cones	101	(3)
Classification of Existence Results for Efficient Points	104	(6)
Some Density and Connectedness Results	110	(5)
Continuity Properties with Respect to a Scalarization Parameter	115	(4)
Well-Posedness of Vector Optimization Problems	119	(3)
Continuity Properties	122	(19)
Continuity Properties of Optimal-Value Multifunctions	122	(13)
Continuity Properties for the Optimal Multifunction in the Case of Moving Cones	135	(3)
Continuity Properties for the Solution Multifunction	138	(3)
Sensitivity of Vector Optimization Problems	141	(13)
Duality	154	(14)
Duality Without Scalarization	155	(4)
Duality by Scalarization	159	(3)
Duality for Approximation Problems	162	(6)
Vector Equilibrium Problems and Related Topics	168	(18)
Vector Equilibrium Problems	169	(2)
General Vector Monotonicity	171	(1)
Existence of Vector Equilibria by Use of the Generalized KKM Lemma	172	(2)
Existence by Scalarization of Vector Equilibrium Problems	174	(3)
Some Knowledge About the Assumptions	177	(3)
Some Particular Cases	180	(3)
Mixed Vector Equilibrium Problems	183	(3)
Applications to Vector Variational Inequalities	186	(11)
Vector Variational-Like Inequalities	186	(2)
Perturbed Vector Variational Inequalities	188	(1)
Hemivariational Inequality Systems	189	(3)
Vector Complementarity Problems	192	(2)
Application to Vector Optimization Problems	194	(2)
Minimax Theorem for Vector-Valued Mappings	196	(1)
Minimal-Point Theorems in Product Spaces and Corresponding Variational Principles	197	(16)
Not Authentic Minimal-Point Theorems	199	(3)
Authentic Minimal-Point Theorems	202	(3)
Minimal-Point Theorems and Gauge Techniques	205	(4)
Minimal-Point Theorems and Cone-Valued Metrics	209	(3)
Fixed Point Theorems of Kirk-Caristi Type	212	(1)
Optimality Conditions	213	(16)
Lagrange Multipliers and Saddle Point Assertions	213	(5)
ε-Saddle Point Assertions	218	(11)
Applications	229	(90)
Approximation Problems	229	(15)
General Approximation Problems	229	(6)
Finite-dimensional Approximation Problems	235	(6)
Lp-Approximation Problems	241	(1)
Example: The Inverse Stefan Problem	242	(2)
Solution Procedures	244	(38)
A Proximal-Point Algorithm for Real-Valued Control Approximation Problems	244	(15)
Computer Programs for the Application of the Proximal-Point Algorithm	259	(4)
An Interactive Algorithm for the Vector Control Approximation Problem	263	(3)
Proximal Algorithms for Vector Equilibrium Problems	266	(11)
Relaxation and Penalization	277	(5)
Location Problems	282	(12)
Formulation of the Problem	282	(3)
An Algorithm for the Multicriteria Location Problem	285	(1)
A Mathematica Program for Solving the Multicriteria Location Problem	286	(1)
Comparison of Alternatives	287	(2)
Application to a Problem of Town Planning	289	(5)
Multicriteria Fractional Programming	294	(9)
Solution Concepts	294	(3)
Generalized Dinkelbach Transformation	297	(3)
Possibilities for a Solution Approach	300	(3)
Multicriteria Control Problems	303	(13)
The Formulation of the Problem	303	(1)
An ε-Minimum Principle for Multicriteria Optimal Control Problems	304	(5)
A Multicriteria Stochastic Control Problem	309	(7)
Stochastic Efficiency in a Set	316	(3)
List of Abbreviations	319	(6)
References	325	(20)
Index	345