Some examples are provided to illustrate these results. This paper is concerned with gap functions of generalized vector variational inequalities gvvi. Google scholar 19 guolin yu, topological properties of henig globally efficient solutions of setvalued optimization problems, numerical algebra, 4 2014. 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. This paper deals with ekelands variational principle for vector optimization problems. Some topics in variational analysis and optimization. We introduce a new efficiency concept which extends and unifies different approximate solution notions introduced in the literature. This paper deals with approximate efficient solutions of vector optimization problems. For further details on vector optimization and setvalued vector optimization, we refer to 22, 29, 32. Optimality conditions for multiobjective optimization. Vector optimization lecture notes in economics and. Decomposition of generalized vector variational inequalities.
The main difficulty here stems from the fact that the perturbation map for such problems is, in general, setvalued. Vector variational inequalities and vector optimization. In the last three decades, the theory of variational analysis provides. Optimality conditions for vector optimization with set.
Fixed point theory, variational analysis, and optimization. Then, by using these relationships and some mild conditions, scalarvalued gap functions for gvvi are. A new variant of ekelands variational principle for set. 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 importance of variational analysis has been proven for problems of multiobjective optimization with singlevalued vectorial and then setvalued objectives. Sufficient conditions for the upper and lower semicontinuity of the. Nonconvex vector optimization of setvalued mappings. An existence result for generalized minty variationallike inequalities and set valued optimization problems is also given. Therefore this relatively new discipline has justifiably attracted a great deal of attention in recent years. By using this principle, we prove the existence of solutions to a vector optimization problem with a setvalued map. The concept of vector variational inequality vvi in finitedimensional spaces. Scalarization approaches for setvalued vector optimization problems and vector variational inequalities. From setvalued optimization problems to setvalued equilibrium problems. By using scalarization approach, scalarvalued variational inequalities of gvvi are introduced.
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. Meanvalue theorems are important and useful tools in nonsmooth analysis. Recently, there has been an increasing interest in the extension of vector optimization to setvalued optimization. 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 and variational analysis find, read and cite all the research you need on researchgate. This principle has been an important tool for nonlinear analysis and optimization theory. The perturbation map is defined as a setvalued map. 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.
Request pdf on jan 1, 2005, guangya chen and others published vector optimization. Characterizations of solution sets of setvalued generalized. Vector optimization setvalued and variational analysis guang. Setvalued and variational analysis vector optimization model has found many important applications in decision making problems such as those in economics. Pdf vector variational inequalities and vector optimization. Boundedness and nonemptiness of solution sets for set. On the vectorial ekelands variational principle and minimal points in product spaces.
Optimality for setvalued optimization in the sense of vector and set. Based on the alternative theorem and some other lemmas, we present necessary optimality conditions and su. Over the past decades, the setvalued vector optimization theory and. Setvalued and variational analysis, lecture notes in economics and mathematical systems. Sensitivity analysis in setvalued optimization and vector variational inequalities. A note on scalarvalued gap functions for generalized. 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. Existence of solutions for vector optimization problems.
Painlevekuratowski convergences for the solution sets of. It is shown that these two conditions are equivalent. Variational inequalities for setvalued vector fields on. 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. The ekeland variational principle for setvalued maps involving. 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. His has published more than seventy papers on setvalued optimization, inverse problems, and variational inequalities. Since setvalued maps subsumes single valued maps, setvalued optimization provides.
In this paper, firstly, a new property of the cone subpreinvex setvalued map involving the generalized contingent epiderivative is obtained. 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. 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. Vector optimization model has found many important applications in decision making. Scalarization approach for approximation of weakly. 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. This includes the more general problems of optimization theory, including topics in setvalued analysis, e. 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. Setvalued optimization problem, setvalued equilibrium problem, vector equilibrium problem, vector variational inequality, normal cone, coderivative, optimality conditions, e cient solutions proper, weakly, strongly, scalarization. Preface to the special issue variational analysis and its applications. Mathematical vector optimization in partially ordered linear spaces, peter. He is a coauthor of setvalued optimization, springer 2015, and coeditor of nonlinear analysis and variational problems, springer 2009. Stability and sensitivity analysis in convex vector.
As an application of this property, a sufficient optimality condition for constrained setvalued optimization problem in the sense of globally proper efficiency is derived. Globally proper efficiency of setvalued optimization and. Sensitivity analysis in setvalued optimization and vector. In this paper stability and sensitivity of the efficient set in convex vector optimization are considered. Global proper efficiency and vector optimization with cone. New algorithms for discrete vector optimization based on the graefyounes method and conemonotone sorting functions. Set optimization and applications the state of the art. Ioffeproximal analysis and approximate subdifferentials. A setvalued ekelands variational principle in vector.
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. The paper deals with the properties of a conic setvalued function defined on the set of all ideal points of vector programming problems. 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. By using a setvalued metric, a setvalued perturbed map, and a coneboundedness concept based on. 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.
Special issue dedicated to professor johannes jahn on the occasion of his 65th birthday. Variational inequalities and vector optimization 2014 hindawi. We consider variational inequality problems for setvalued vector fields on general riemannian manifolds. 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. An alternative theorem for setvalued maps via set relations and its application. Finally, we establish the relations between the globally proper efficiency of the. In this paper, we establish a farkasminkowski type alternative theorem under the assumption of nearly semiconvexlike setvalued maps.
Vector optimization setvalued and variational analysis. 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. Finally, an existence result of generalized weakly efficient solutions for vector optimization problem involving a subdifferentiable and preinvex setvalued mapping is established by exploiting the existence of a solution for the weak formulation of the generalized stampacchia vector variationallike inequality via a fankkm lemma. Initially, secondorder necessary optimality conditions and sufficient optimality conditions in terms of hadamard type derivatives for the unconstrained scalar optimization problem. 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. Optimality conditions for vector optimisation with set. Stability analysis for setvalued vector mixed variational. 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. 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 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. 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. The first chapter provides basic notations and results from the areas of convex analysis, functional analysis, setvalued analysis and fixedpoint.