Therefore, we introduce some local conditions and we prove that these conditions are useful mainly in the study of borwein proper minima for the considered problems. Padhan, optimality conditions and duality results for nondifferentiable interval optimization problems, j. Optimality for setvalued optimization in the sense of vector and set criteria. Pdf generalized constraint qualifications and optimality. The aim of this paper was to provide optimality conditions for setvalued optimisation problems with respect to the set less order relation. Set optimization and applications the state of the art. Moreover we present the new concept of isolated minimizer for set valued optimization. Necessary and sufficient conditions for the existence of solutions are shown for setvalued maps under generalized convexity assumptions and with the notion. Optimality conditions for a class of mathematical programs. Sufficient conditions for the existence of solutions are shown for setvalued maps under generalized quasiconvexity assumptions. A unified approach and optimality conditions for approximate. Subdifferential necessary conditions for extremal solutions to setvalued optimization problems with equilibrium constraints, bao q. We extend some of the existing concepts to general spaces and cones using set relations. Setvalued optimization ebook por christiane tammer.
Optimality conditions for vector optimization with set. Set valued optimization is a vibrant and expanding branch of mathematics that deals with optimization problems where the. In this work, we explore the notion of strongly convex functions of order. Set valued optimization is a vibrant and expanding branch of mathematics that deals with optimization problems where the objective map andor the constraints maps are set valued maps acting between certain spaces. For the case of wminimizers some comparison with existing results is done. Optimality conditions of type kkt for optimization problem with interval valued objective function via generalized derivative article pdf available in fuzzy optimization and decision making 123. Convexity of the multifunction and the domain is not required. Optimality conditions for vector optimization with setvalued. A unified necessary and sufficient optimality condition is derived in terms of the generalized contingent epiderivative. On optimality conditions and duality for nondifferentiable. Several stationarity concepts, based on a piecewise smooth formulation, are presented and compared.
The third part is on multivalued setvalued optimization. A necessary and sufficient optimality condition for. First order optimality condition for constrained setvalued. We consider both global and local conditions for optimization problems governed by setvalued maps. In this paper optimality conditions are studied for set valued maps with set optimization. Li 6, under the assumption of conesubconvexlikeness of setvalued maps, established optimality conditions for setvalued vector optimization by using the alternative theorem in ordered linear topological spaces. We study mathematical programs with complementarity constraints. Based on the concept of an epiderivative for a setvalued map introduced in j.
The new topics studied include the formulation of optimality conditions using. In this work, we use a notion of convexificator 25 together with the support function 3, 4, 15, 16, 41 to establish necessary optimality conditions for set valued bilevel optimization problems. In the last decade vector optimization has been extended to problems with set valued maps see 11. Optimality conditions in terms of bouligand derivatives for pareto efficiency in set valued optimization m durea, r strugariu optimization letters 5 1, 141151, 2011. In this paper optimality conditions are studied for setvalued maps with set optimization. In this bibliography main directions of research as well as main fields of applications of bilevel programming problems and mathematical programs with equilibrium constraints are summarized. Rd ris lipschitz continuous around the point of interest and rd. Optimality conditions for vector optimization with set valued maps. Journal of industrial and management optimization 7. Kepiderivatives for setvalued functions and optimization. Following the same ideas, amahroq and gadhi 6, 7 have established optimality conditions to some optimization problems under setvalued mapping constraints. Setvalued numerical analysis and optimal control in. Exploiting different tangent cones, many derivatives for setvalued functions have been introduced and considered to study optimality. Optimality conditions for setvalued maps with set optimization.
Global and local optimality conditions in setvalued. In 22, khan and tammer gave new secondorder optimality conditions in setvalued optimization. Pdf optimality conditions of type kkt for optimization. Strugariu, optimality conditions in terms of bouligand derivatives for pareto efficiency in set valued optimization, optimization letters, 5 2011, 141151. Vector optimization, setvalued optimization, firstorder optimality conditions. In this paper, we discuss the connection between concepts of robustness for multiobjective optimization problems and set order relations. Optimality conditions for constrained optimization problems robert m. The chapters were written by outstanding experts in the areas of bilevel programming, mathematical programs with equilibrium or complementarity constraints mpec, and setvalued optimization problems. Durea, global and local optimality conditions in setvalued optimization problems, international journal of mathematics and mathematical sciences, 11 2005, 16931711. This paper mainly studies the optimality conditions for a class of pessimistic trilevel optimization problem, of which middlelevel is a pessimistic problem.
This paper mainly studies the optimality conditions for a class of trilevel optimization problem, of which all levels are nonlinear programs. The existence of weak subgradients of setvalued maps is proved, and a sufficient optimality condition of setvalued optimization problems is obtained in terms of weak subgradients. In this paper we study optimization problems with variational inequality constraints in finite dimensional spaces. Quasirelative interiors for graphs of convex setvalued mappings optimization letters, in press. Mordukhovich, we derive 1storder necessary optimality conditions.
A necessary and sufficient optimality condition for bilevel. The maximum principle for the nonlinear stochastic optimal. To dualspace theory of setvalued optimization, bao q. Necessary optimality conditions for optimization problems. Optimality conditions are studied for setvalued maps with set optimization. Mathematical programs with complementarity constraints.
Necessary optimality conditions in a problem of terminal control with nonfunctional constraints journal doklady akademii nauk bssr 20 1976, no. Optimality conditions for setvalued optimisation problems. Extended pareto optimality in multiobjective problem, bao q. Optimality conditions for pessimistic trilevel optimization. Optimality for setvalued optimization in the sense of vector. Since setvalued maps subsumes single valued maps, setvalued optimization provides an important extension and unification of the. Kuhntucker type necessary optimality conditions involving coderivatives are given under certain constraint qualifications including one that ensures nonexistence of nontrivial abnormal multipliers. Optimality conditions for optimization problems with. Necessary and sufficient conditions for the existence of solutions are shown for set valued maps under generalized convexity assumptions and with the notion. We consider both global and local conditions for optimization problems governed by set valued maps. Then a farkasminkowski type alternative theorem is shown under. Strugariu, optimality conditions in terms of bouligand derivatives for pareto efficiency in setvalued optimization, optimization letters, 5 2011, 141151. The existence of weak subgradients of set valued maps is proved, and a sufficient optimality condition of set valued optimization problems is obtained in terms of weak subgradients.
We rstly translate this problem into an auxiliary pessimistic bilevel optimization problem, by applying kkt approach for the lower level problem. Optimality conditions for a nonconvex setvalued optimization problem. Goal of this research to provide methods that can be implemented to obtain numerical results and optimality conditions for locating set solutions of setvalued optimization problems. Secondorder dini setvalued directional derivative in c 1.
These notions are investigated and appear when establishing firstorder necessary and sufficient optimality conditions derived in terms of a dini type derivative for set valued maps. Optimality conditions for proper efficient solutions of. Optimization online optimality conditions for vector. Durea, variational inclusions for contingent derivative of set valued maps, journal of mathematical analysis and applications, 292 2004, 3563.
Buy set optimization and applications the state of the art. Focus is also on the difficulties arising from nonuniqueness of lowerlevel optimal solutions and on optimality conditions. Optimality conditions for constrained optimization problems. This book contains the latest advances in variational analysis and set vector optimization, including uncertain optimization, optimal control and bilevel optimization. Optimality conditions in terms of bouligand derivatives for pareto efficiency in setvalued optimization. Necessary conditions are given in terms of derivative and contingent derivative. Optimality conditions for several types of efficient. This new eld of research called set optimization seems to have important applications, and presents a new di culty. Based on near convexity, we introduce the concepts of nearly convexlike setvalued maps and nearly semiconvexlike setvalued maps, give some charaterizations of them, and investigate the relationships between them.
Sparse optimization for inverse problems in atmospheric modelling. Optimization methods and software 31 3, 535561, 2016. They presented an extension of the wellknown dubovitskimilutin approach to setvalued. In the last decade vector optimization has been extended to problems with setvalued maps see 11. Generalized constraint qualifications and optimality conditions for setvalued optimization problems article pdf available in journal of mathematical analysis and applications 2652. We firstly transform this problem into an auxiliary bilevel optimization problem by applying kkt approach to the lowerlevel problem. Recent developments concerning scalarization techniques, necessary and sufficient optimality conditions and. Siam journal on optimization society for industrial and. Therefore, we introduce some local conditions and we prove that these conditions are useful mainly in the study of borwein proper minima for the.
Frechet subdifferential calculus and optimality conditions in. The role of the iminimizers, which seems to be a new concept in set valued optimization, is underlined. Fortunately, the lipschitz property of a setvalued mapping is. The relationship between multiobjective robustness. Optimality conditions for weak and firm efficiency in set valued optimization m durea journal of mathematical analysis and applications 344 2, 10181028, 2008. Improved optimality conditions for scalar, vector and set. Optimality conditions for weak and firm efficiency in setvalued optimization article in journal of mathematical analysis and applications 3442. The role of the iminimizers, which seems to be a new concept in setvalued optimization, is underlined.
Based on near convexity, we introduce the concepts of nearly convexlike set valued maps and nearly semiconvexlike set valued maps, give some charaterizations of them, and investigate the relationships between them. The topics range from more conventional approaches that look for minimalmaximal elements with respect to vector orders or set relations, to the new completelattice approach that comprises a coherent solution concept for set optimization problems, along with existence results, duality theorems, optimality conditions, variational inequalities. Optimality conditions for weak and firm efficiency in set. This multifunction is the differentiability notion applied to the problem. By virtue of the higherorder derivatives introduced in aubin and frankowska, setvalued analysis, birkhauser, boston, 1990 higherorder necessary and sufficient optimality conditions are obtained for a setvalued optimization problem whose constraint condition is determined by a fixed set. For global conditions, we present a comparative study and then we impose the weaker ones to obtain optimality conditions. Necessary and sufficient conditions for setvalued maps. Fortunately, the lipschitz property of a set valued mapping is conserved for its support function. Freund february, 2004 1 2004 massachusetts institute of technology. Optimality for set valued optimization in the sense of vector and set criteria.
The maximum principle for the nonlinear stochastic optimal control problem of switching systems. Rufianlizana, optimality conditions of type kkt for optimization problem with intervalvalued object function via. Li 6, under the assumption of conesubconvexlikeness of set valued maps, established optimality conditions for set valued vector optimization by using the alternative theorem in ordered linear topological spaces. Improved optimality conditions for scalar, vector and setvalued optimization problems doctoral thesis summary supervisor. Optimality conditions of setvalued optimization problem. 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. Durea, variational inclusions for contingent derivative of setvalued maps, journal of mathematical analysis and applications, 292 2004, 3563. Setvalued numerical analysis and optimal control r.
Recent developments concerning scalarization techniques, necessary and sufficient optimality conditions and duality statements are. For this purpose, we use a slightly modified demyanov difference in order to introduce a sort of directional derivative for setvalued maps, which allows us to derive optimality conditions. Optimality conditions for a nonconvex setvalued optimization. Vector optimization, set valued optimization, firstorder optimality conditions. Because interval valued programming problem is used to tackle interval uncertainty that appears in many mathematical or computer models of some deterministic realworld phenomena, this paper considers a nondifferentiable interval valued optimization problem in which objective and all constraint functions are interval valued functions, and the involved endpoint functions in interval valued. First order optimality conditions in setvalued optimization. Constraint qualifications and necessary optimality conditions for optimization problems. The main goal of the paper is to address a general concept of kepiderivative and to employ it to develop a quite general scheme for necesary optimality conditions in setvalued problems. In terms of the secondorder dini setvalued directional derivative, secondorder necessary conditions a point x 0 to be a weakly efficient point and secondorder sufficient conditions x 0 to be an isolated minimizer of order two are established. Mathematics and computer science college of arts and.
On setvalued optimization problems with variable ordering structure. Optimization problems with complementarity constraints are closely related to opti. Durea, global and local optimality conditions in set valued optimization problems, international journal of mathematics and mathematical sciences, 11 2005, 16931711. The concepts are related to stationarity conditions for certain smooth programs as well as to stationarity concepts for a nonsmooth exact penalty function. Optimality conditions of setvalued optimization problem involving relative algebraic interior in ordered linear spaces zhiang zhoua, xinmin yangb and jianwen pengc1 adepartment of applied mathematics, chongqing university of technology, chongqing 400054, p. Furthermore, we derive new concepts of robustness for multiobjective optimization problems. Optimization and interval analysis for technological applications author. Dec 18, 2007 this paper deals with higherorder optimality conditions of setvalued optimization problems.
Optimization and interval analysis for technological. Optimality requirements are established for continuous selections using directional derivatives. Optimization and interval analysis for technological applications. The imposed constraint qualification is studied in detail and compared with other conditions arising in this context. 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. Using the generalized differential calculus for nonsmooth and setvalued mappings due to b. Higherorder optimality conditions for setvalued optimization. Optimality for setvalued optimization in the sense of. Solving a continuous multifacility location problem by dc programs journal of optimization methods and software, submitted a. Optimality conditions for various e cient solutions involving coderivatives.
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