1,721,208 research outputs found
Some topological properties of solution sets in partially ordered set optimization
No full textIn this paper, we study some topological properties, in particular, arcwise connectedness and connectedness of solution sets in set optimization, where the acting space is equipped with partial set order relations. We obtain continuity, generalized convexity, and natural quasi arcwise connectedness of the nonlinear scalarization function and use them to study some topological properties and convergence of efficient and weak efficient solution sets in partially ordered set optimization. We also employ derived results to vector-valued game theory with uncertainty. </p
Some properties of generalized oriented distance function and their applications to set optimization problems
No full textIn this paper, we study several interesting basic properties of generalized oriented distance function with respect to co-radiant sets or free disposal sets, which are more general than a cone and play an important role to study quasi-minimal solutions of set optimization problems. In particular, we deal with some special properties, namely, translation property, subadditivity and monotonicity, by using co-radiant sets. Moreover, we investigate several kinds of monotonicity properties by means of nonconvex free disposal sets. As an application, we study some optimality conditions for quasi-minimal solutions of set optimization problems by using generalized oriented distance function. At the end, we give an existence theorem for cone saddle-point for set-valued maps. Several examples are given to verify the validity and effectiveness of the derived results.</p
Connectedness of the solution sets in generalized semi-infinite set optimization
No full textWe first establish sufficient conditions for the arcwise connectedness of the image of the constraint set map and for the upper semi-continuity of the constraint set map. These results, together with scalarization techniques, are further used to establish the connectedness of the solution sets of generalized semi-infinite set optimization problems. An application to vector-valued game theory with uncertainty is given.</p
Levitin-polyak well-posedness for set optimization problems
No full textIn this paper, we study different kinds of Levitin-Polyak well-posedness for set optimization problems and their relationships with respect to the set order relations defined by Minkowski difference on the family of bounded sets. Furthermore, by using the Kuratowski measure of noncompactness, we give some characterizations of Levitin-Polyak well-posedness for set optimization problems. Moreover, we establish the relationship between stability and LP well-posedness of set optimization problem by defining approximating solution maps. Several examples are given in support of concepts and results of this paper.</p
Characterizations of multiobjective robustness via oriented distance function and image space analysis
No full textIn this paper, we characterize different kinds of multiobjective robustness concepts via the well-known oriented distance function. By using characterizations of several set relations via the oriented distance function, together with the help of image space analysis, we construct some suitable subsets of the scalarization image space to obtain equivalent characterizations for various robust solutions for uncertain multiobjective optimization problems based on a set approach.</p
Convergence of the solution sets for set optimization problems
No full textIn this paper, we present the stability analysis of solution sets for set optimization problems with respect to the set order relation defined by means of Minkowski difference. We introduce the concepts of weak/weak# locally Lipschitz continuity and the concepts of ml-quasiconnectedness and strictly ml-quasiconnectedness for set-valued mappings. By using these concepts, we study the Painlevé-Kuratowski convergence of the solution sets for perturbed set optimization problems. Several examples are given to illustrate our results.</p
Ekeland’s variational principle with weighted set order relations
No full textThe main results of the paper are a minimal element theorem and an Ekeland-type variational principle for set-valued maps whose values are compared by means of a weighted set order relation. This relation is a mixture of a lower and an upper set relation which form the building block for modern approaches to set-valued optimization. The proofs rely on nonlinear scalarization functions which admit to apply the extended Brézis–Browder theorem. Moreover, Caristi’s fixed point theorem and Takahashi’s minimization theorem for set-valued maps based on the weighted set order relation are obtained and the equivalences among all these results is verified. An application to generalized intervals is given which leads to a clear interpretation of the weighted set order relation and versions of Ekeland’s principle which might be useful in (computational) interval mathematics.</p
Minimal element theorems and Ekeland's variational principle with new set order relations
No full textBy using scalarization functions, we study minimal element theorem, Ekeland's variational principle, Caristi's fixed point theorem, Takahashi's minimization theorem under the set order relations on the family of sets defined by means of Minkowski difference. We also give some characterizations of set order relations in terms of oriented distance function.</p
Set order relations, set optimization, and Ekeland’s variational principle
No full textThis chapter provides a brief survey on different kinds of set order relations which are used to compare the objective values of set-valued maps and play a key role to study set optimization problems. The solution concepts of set optimization problems and their relationships with respect to different kinds of set order relations are provided. The nonlinear scalarization functions for vector-valued maps as well as for set-valued maps are very useful to study the optimality solutions of vector optimization/set optimization problems. A survey of such nonlinear scalarization functions for vector-valued maps/set-valued maps is given. We give some new results on the existence of optimal solutions of set optimization problems. In the end, we gather some recent results, namely, Ekeland’s variational principle and some equivalent variational principle for set-valued maps with respect to different kinds of set order relations.</p
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