Synopses & Reviews
The book deals with the mathematical theory of vector variational inequalities with special reference to equilibrium problems. Such models have been introduced recently to study new problems from mechanics, structural engineering, networks, and industrial management, and to revisit old ones. The common feature of these problems is that given by the presence of concurrent objectives and by the difficulty of identifying a global functional (like energy) to be extremized. The vector variational inequalities have the advantage of both the variational ones and vector optimization which are found as special cases. Among several applications, the equilibrium flows on a network receive special attention. Audience: The book is addressed to academic researchers as well as industrial ones, in the fields of mathematics, engineering, mathematical programming, control theory, operations research, computer science, and economics.
Synopsis
In the fifties and sixties, several real problems, old and new, especially in Physics, Mechanics, Fluidodynamics, Structural Engi- neering, have shown the need of new mathematical models for study- ing the equilibrium of a system. This has led to the formulation of Variational Inequalities (by G. Stampacchia), and to the develop- ment of Complementarity Systems (by W.S. Dorn, G.B. Dantzig, R.W. Cottle, O.L. Mangasarian et al.) with important applications in the elasto-plastic field (initiated by G. Maier). The great advan- tage of these models is that the equilibrium is not necessarily the extremum of functional, like energy, so that no such functional must be supposed to exist. In the same decades, in some fields like Control Theory, Net- works, Industrial Systems, Logistics, Management Science, there has been a strong request of mathmatical models for optimizing situa- tions where there are concurrent objectives, so that Vector Optimiza- tion (initiated by W. Pareto) has received new impetus. With regard to equilibrium problems, Vector Optimization has the above - mentioned drawback of being obliged to assume the exis- tence of a (vector) functional. Therefore, at the end of the seventies the study of Vector Variational Inequalities began with the scope of exploiting the advantages of both variational and vector models. This volume puts together most of the recent mathematical results in Vector Variational Inequalities with the purpose of contributing to further research.
Table of Contents
Preface. Vector Equilibrium Problems and Vector Variational Inequalities; A.H. Ansari. Generalized Vector Variational-Like Inequalities and their Scalarization; A.H. Ansari, et al. Existence of Solutions for Generalized Vector Variational-Like Inequalities; S.-S. Chang, et al. On Gap Functions for Vector Variational Inequalities; G.-Y. Chen, et al. Existence of Solutions for Vector Variational Inequalities; G.-Y. Chen, S.-H. Hou. On the Existence of Solutions to Vector Complementarity Problems. Vector Variational Inequalities and Modelling of a Continuum Traffic Equilibrium Problem; P. Daniele, A. Maugeri. Generalized Vector Variational-Like Inequalities without Monotonicity; X.P. Ding, E. Tarafdar. Generalized Vector Variational-Like Inequalities with Cx-eta-Pseudomonotone Set-Valued Mappings; X.P. Ding, E. Tarafdar. A Vector Variational-Like Inequality for Compact Acyclic Multifunctions and its Applications; J. Fu. On the Theory of Vector Optimization and Variational Inequalities. Image Space Analysis and Separation; F. Giannessi, et al. Scalarization Methods for Vector Variational Inequality; C.J. Goh, X.Q. Yang. Super Efficiency for a Vector Equilibrium in Locally Convex Topological Vector Spaces; X.H. Gong, et al. The Existence of Essentially Connected Components of Solutions for Variational Inequalities; G. Isac, G.X.Z. Yuan. Existence of Solutions for Vector Saddle-Point Problems; K.R. Kazmi. Vector Variational Inequality as a Tool for Studying Vector Optimization Problems; G.M. Lee, et al. Vector Variational Inequalities in a Hausdorff Topological Vector Space; G.M. Lee, S. Kum. Vector Ekeland Variational Principle; S.J. Li, et al. Convergence of Approximate Solutions and Values in Parametric Vector Optimization; P. Loridan, J. Morgan. On Minty Vector Variational Inequality; G. Mastroeni. Generalized Vector Variational-Like Inequalities; L. Qun. On Vector Complementarity Systems and Vector Variational Inequalities; T. Rapcsák. Generalized Vector Variational Inequalities; W. Song. Vector Equilibrium Problems with Set-Valued Mappings; W. Song. On Some Equivalent Conditions of Vector Variational Inequalities; X.Q. Yang. On Inverse Vector Variational Inequalities; X.Q. Yang, G.-Y. Chen. Vector Variational Inequalities, Vector Equilibrium Flow and Vector Optimization; X.Q. Yang, C.-J. Goh. On Monotone and Strongly Monotone Vector Variational Inequalities; N.D. Yen, G.M. Lee. Connectedness and Stability of the Solution Sets in Linear Fractional Vector Optimization Problems; N.D. Yen, T.D. Phuong. Vector Variational Inequality and Implicit Vector Complementarity Problems; H. Yin, C. Xu. References on Vector Variational Inequalities. Subject Index. Contributors.