Variational inequalities and vector optimization 2014 hindawi. An alternative theorem for setvalued maps via set relations and its application. Optimality for setvalued optimization in the sense of vector and set. Some topics in variational analysis and optimization. Based on the alternative theorem and some other lemmas, we present necessary optimality conditions and su. There is no comparable volume on the market, making the book an invaluable resource for researchers working in vector optimization and multicriteria decisionmaking, mathematical finance and economics as well as setvalued variational analysis. Nonconvex vector optimization of setvalued mappings. Meanvalue theorems are important and useful tools in nonsmooth analysis. The results here about the continuity and derivability of this conic setvalued map, can be used to get information about the sensitivity of the problem and the stability of the order associated to every ideal point. Sensitivity analysis in setvalued optimization and vector. Then, by using these relationships and some mild conditions, scalarvalued gap functions for gvvi are. By using this principle, we prove the existence of solutions to a vector optimization problem with a setvalued map. It associates with each parameter vector the set of all minimal points of the parametrized feasible set with respect to an ordering cone in the objective space.
Along with the development of vector optimization and setvalued optimization, the vector variational principle introduced by nemeth 1980 has been an interesting topic in the last decade. By using scalarization approach, scalarvalued variational inequalities of gvvi are introduced. Request pdf on jan 1, 2005, guangya chen and others published vector optimization. Sufficient conditions for the upper and lower semicontinuity of the. It is shown that these two conditions are equivalent. Setvalued optimization is a vibrant and expanding branch of mathematics that deals with optimization problems where the objective map andor the constraints maps are setvalued maps acting between certain spaces. Vector optimization setvalued and variational analysis. As a bridge between different areas of optimization, the theory of setvalued optimization problems has wide applications in differential inclusion, variational inequality, optimal control, game theory, economic equilibrium problem, decision making, etc. Based on the alternative theorem and some other lemmas, we establish necessary optimality conditions for setvalued vector optimisation problems with extended inequality constraints in a sense of weak eminimisers. He is a coauthor of setvalued optimization, springer 2015, and coeditor of nonlinear analysis and variational problems, springer 2009. Vector variational inequalities and vector optimization. In this paper, we establish a farkasminkowski type alternative theorem under the assumption of nearly semiconvexlike setvalued maps. Optimality conditions for multiobjective optimization.
Some equivalence results among setvalued scalar optimization problems, setvalued scalar quasioptimization problems, setvalued vector optimization problems and setvalued weak vector optimization problems are established under the convexity assumption of objective functions. Mathematical vector optimization in partially ordered linear spaces, peter. Since setvalued maps subsumes single valued maps, setvalued optimization provides. From setvalued optimization problems to setvalued equilibrium problems. Characterizations of solution sets of setvalued generalized. The aim of this paper is to study a link between the optimal points of vector optimization problems governed by setvalued maps and the solutions of some related generalized variational inequalities. In this paper, some characterizations for the solution sets of a class of setvalued vector mixed variational inequalities to be nonempty and bounded are presented in real reflexive banach spaces.
Since setvalued maps subsumes single valued maps, setvalued optimization provides an important extension and unification of the scalar as well as the vector optimization problems. Global proper efficiency and vector optimization with cone. Painlevekuratowski convergences for the solution sets of. A setvalued ekelands variational principle in vector. The main difficulty here stems from the fact that the perturbation map for such problems is, in general, setvalued. On the other hand, the differentiability issues of the perturbation map for vector optimization problems and set optimization problems are rather involved and they require modern tools from variational analysis. Preface to the special issue variational analysis and its applications. Some examples are provided to illustrate these results. Boundedness and nonemptiness of solution sets for set. We consider variational inequality problems for setvalued vector fields on general riemannian manifolds. Existence of solutions for vector optimization problems.
Read optimality conditions for multiobjective optimization problem constrained by parameterized variational inequalities, setvalued and variational analysis on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at. The ekeland variational principle for setvalued maps involving. Fixed point theory, variational analysis, and optimization is a beneficial resource for the research and study of nonlinear analysis, optimization theory, variational inequalities, and mathematical economics. By using a setvalued metric, a setvalued perturbed map, and a coneboundedness concept based on. Fixed point theory, variational analysis, and optimization. Variational inequalities for setvalued vector fields on. The concept of vector variational inequality vvi in finitedimensional spaces. In the last three decades, the theory of variational analysis provides. Initially, secondorder necessary optimality conditions and sufficient optimality conditions in terms of hadamard type derivatives for the unconstrained scalar optimization problem. The first chapter provides basic notations and results from the areas of convex analysis, functional analysis, setvalued analysis and fixedpoint theory for setvalued maps, as well as a brief introduction to variational inequalities and equilibrium problems. This paper is concerned with gap functions of generalized vector variational inequalities gvvi. Since the concept of vector variational inequality vvi was introduced by giannessi in 1980, many important results on various kinds of vector variational inequality problems have been established, such as existence of solutions, relations with vector optimization, stability of solution set maps, gap function, and duality theories see, e. Set optimization and applications the state of the art.
Optimality conditions for vector optimisation with set. Pdf vector variational inequalities and vector optimization. Ioffeproximal analysis and approximate subdifferentials. Vector optimization model has found many important applications in decision making. The paper deals with the properties of a conic setvalued function defined on the set of all ideal points of vector programming problems. Scalarization approach for approximation of weakly. Conic setvalued maps in vector optimization, setvalued. Google scholar 19 guolin yu, topological properties of henig globally efficient solutions of setvalued optimization problems, numerical algebra, 4 2014. This principle has been an important tool for nonlinear analysis and optimization theory.
Finally, we establish the relations between the globally proper efficiency of the. Optimality conditions for vector optimization with set. In this paper, firstly, a new property of the cone subpreinvex setvalued map involving the generalized contingent epiderivative is obtained. We define the generalized efficient solution which is more general than the weakly efficient solution for vector optimization problems, and prove the existence of the generalized efficient solution for nondifferentiable vector optimization problems by using vector variationallike. Some relationships between the solutions of gvvi and its scalarized versions are established. Recently, there has been an increasing interest in the extension of vector optimization to setvalued optimization. Vector optimization lecture notes in economics and. Globally proper efficiency of setvalued optimization and. Therefore this relatively new discipline has justifiably attracted a great deal of attention in recent years. The importance of variational analysis has been proven for problems of multiobjective optimization with singlevalued vectorial and then setvalued objectives.
Stability and sensitivity analysis in convex vector. In mathematics, the term variational analysis usually denotes the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory. By using a setvalued metric, a setvalued perturbed map, and a coneboundedness concept based on scalarization, we introduce an original approach to extending the wellknown scalar ekelands principle to. Setvalued and variational analysis, lecture notes in economics and mathematical systems. The first chapter provides basic notations and results from the areas of convex analysis, functional analysis, setvalued analysis and fixedpoint. We obtain necessary and sufficient conditions via nonlinear scalarization, which allow us to study this new class of approximate solutions in a general framework. The existence results of the solution, convexity of the solution set, and the convergence property of the proximal point algorithm for the variational inequality problems for setvalued mappings on riemannian manifolds are established. A new variant of ekelands variational principle for set. Most of the usual calculus rules, from chain and sum rules to rules for unions, intersections, products and other operations on mappings, are established.
Setvalued optimization problem, setvalued equilibrium problem, vector equilibrium problem, vector variational inequality, normal cone, coderivative, optimality conditions, e cient solutions proper, weakly, strongly, scalarization. An existence result for generalized minty variationallike inequalities and set valued optimization problems is also given. As an application of this property, a sufficient optimality condition for constrained setvalued optimization problem in the sense of globally proper efficiency is derived. Decomposition of generalized vector variational inequalities. Gap functions and existence of solutions to set valued vector variational inequalities, journal of. Vector optimization setvalued and variational analysis guang.
For further details on vector optimization and setvalued vector optimization, we refer to 22, 29, 32. This paper deals with approximate efficient solutions of vector optimization problems. By using a setvalued metric, a setvalued perturbed map, and a coneboundedness concept based on scalarization, we introduce an original approach to extending the wellknown scalar ekelands principle to vector valued maps. An equivalence relation between the solution sets of the vector mixed variational inequalities and the weakly efficient solution sets of the vector optimization problems is shown under some suitable. New algorithms for discrete vector optimization based on the graefyounes method and conemonotone sorting functions. Over the past decades, the setvalued vector optimization theory and. We introduce a new efficiency concept which extends and unifies different approximate solution notions introduced in the literature. A note on scalarvalued gap functions for generalized. Fixed point theory, variational analysis, and optimization not only covers three vital branches of nonlinear analysisfixed point theory, variational inequalities, and vector optimizationbut also explains the connections between them, enabling the study of a general form of variational inequality problems related to the optimality conditions involving differentiable or. This paper deals with ekelands variational principle for vector optimization problems. We develop elements of calculus of variational sets for setvalued mappings, which were recently introduced in khanh and tuan 2008, to replace generalized derivatives in establishing optimality conditions in nonsmooth optimization. This includes the more general problems of optimization theory, including topics in setvalued analysis, e.
Scalarization approaches for setvalued vector optimization problems and vector variational inequalities. Setvalued and variational analysis find, read and cite all the research you need on researchgate. Special issue dedicated to professor johannes jahn on the occasion of his 65th birthday. His has published more than seventy papers on setvalued optimization, inverse problems, and variational inequalities. Setvalued and variational analysis vector optimization model has found many important applications in decision making problems such as those in economics. Sensitivity analysis in setvalued optimization and vector variational inequalities. Stability analysis for setvalued vector mixed variational. In this paper stability and sensitivity of the efficient set in convex vector optimization are considered. The perturbation map is defined as a setvalued map.
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