Solutions Of Set-valued Optimization Problems Research Articles

Approximate Weak Minimal Solutions of Set-Valued Optimization Problems

Approximate Weak Minimal Solutions of Set-Valued Optimization Problems

New Properties of Higher-Order Radial Sets and Higher-Order Radial Derivatives and Applications to Optimality Conditions

: Several new properties are obtained for higher-order upper radial sets and corresponding derivatives introduced in [J. Glob. Optim. DOI 10.1007/s10898-012-9861- z]. Higher-order sufficient optimality conditions and higher-order necessary optimality conditions are established for weak efficient solutions of set-valued optimization problems. Several corollaries are provided to show that obtained results include a few existing ones. In addition, one removes deficiencies contained of two results in [J. Glob. Optim. DOI 10.1007/s10898-012-9861-z].

On Set Containment Characterizations for Sets Described by Set-Valued Maps with Applications

In this paper, dual characterizations of the containment of two sets involving convex set-valued maps are investigated. These results are expressed in terms of the epigraph of a conjugate function of infima associated with corresponding set-valued maps. As an application, we establish characterizations of weak and proper efficient solutions of set-valued optimization problems in the sense of vector criteria.

Scalarized solutions of set-valued optimization problems and generalized variational-like inequalities

In this paper by using the scalariation method we introduced the concept of relaxed K-preinvex set-valued maps and obtain some equivalence results of them in terms of normal subdifferential. Also, we consider generalized Minty variational-like inequalities and show that the set of solutions is equal to scalarized set-valued optimization problems?s solutions under generalized relaxed convexity assumptions.

Approximate quasi efficiency of set-valued optimization problems via weak subdifferential

This paper deals with the approximate quasi efficient solutions of set-valued optimization problems. We establish the necessary and sufficient optimality conditions for \(\epsilon e\)-quasi efficiency via weak subdifferential under some approximate cone convexity assumptions. We also study duality results of Wolfe and Mond-Weir types for Geoffrion \(\epsilon e\)-quasi efficiency of finite dimensional set-valued optimization problems.

Scalarization approach for approximation of weakly efficient solutions of set-valued optimization problems

In this paper, by using the scalarization method and normal subdifferential for set-valued maps, we consider an extension of Minty variational-like inequalities and obtain some relations between their solutions and set-valued optimization problems. An existence result for generalized Minty variational-like inequalities and set-valued optimization problems is also given. Moreover, the concept of approximate efficient solutions due to Kutateladze is investigated and by the Tammer–Weidner nonlinear functional, we characterize them for cone constrained set-valued optimization problems.

Scalarization of $$\epsilon $$ ϵ -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces

In this paper, we investigate the scalarization of $$\epsilon $$ ∈ -super efficient solutions of set-valued optimization problems in real ordered linear spaces. First, in real ordered linear spaces, under the assumption of generalized cone subconvexlikeness of set-valued maps, a dual decomposition theorem is established in the sense of $$\epsilon $$ ∈ -super efficiency. Second, as an application of the dual decomposition theorem, a linear scalarization theorem is given. Finally, without any convexity assumption, a nonlinear scalarization theorem characterized by the seminorm is obtained.

Scalarization in set optimization with solid and nonsolid ordering cones

This paper focuses on characterizations via scalarization of several kinds of minimal solutions of set-valued optimization problems, where the objective values are compared through relations between sets (set optimization). For this aim we follow an axiomatic approach based on general order representation and order preservation properties, which works in any abstract set ordered by a quasi order (i.e., reflexive and transitive) relation. Then, following this approach, we study a recent Gerstewitz scalarization mapping for set-valued optimization problems with $$K$$ K -proper sets and a solid ordering cone $$K$$ K . In particular we show a dual minimax reformulation of this scalarization. Moreover, in the setting of normed spaces ordered by non necessarily solid ordering cones, we introduce a new scalarization functional based on the so-called oriented distance. Using these scalarization mappings, we obtain necessary and sufficient optimality conditions in set optimization. Finally, whenever the ordering cone is solid, by considering suitable generalized Chebyshev norms with appropriate parameters, we show that the three scalarizations studied in the present work are coincident.

On approximate solutions in set-valued optimization problems

In this paper we introduce several concepts of approximate solutions of set-valued optimization problems with vector and set optimization. We prove existence results and necessary and sufficient conditions by using limit sets.

Higher-Order Scalarization Theorems for Strictly Efficient Solutions of Set-Valued Optimization Problems

In this paper,higher-order derivatives type scalarization theorems are discussed for strictly efficient solutions of set-valued optimization problems.Firstly,we obtain a generalized higherorder derivatives type necessary optimality condition for strictly efficient solutions of a set-valued optimization problem.Secondly,scalarization necessary conditions and sufficient conditions are derived for strictly efficient solutions of set-valued optimization problems by employing generalized higher-order epiderivatives.

Well-Posedness for Tightly Proper Efficiency in Set-Valued Optimization Problem

In this paper, a characterization of tightly properly efficient solutions of set-valued optimization problem is obtained. The concept of the well-posedness for a special scalar problem is linked with the tightly properly efficient solutions of set-valued optimization problem.

Strict approximate solutions in set-valued optimization with applications to the approximate Ekeland variational principle

This work deals with strict solutions of set-valued optimization problems under the set optimality criterion. In this context, we introduce a new approximate solution concept and we obtain several properties of these solutions when the error is fixed and also for their limit behavior when the error tends to zero. Then we prove a general existence result, which is applied to obtain approximate Ekeland variational principles.

On solutions of set-valued optimization problems

We consider two criteria of a solution associated with a set-valued optimization problem, a vector criterion and a set criterion. We show how solutions of a vector type can help to find solutions of a set type and reciprocally. As an application, we obtain a sufficient condition for the existence of solutions of a set type via vector optimization.

Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization

In this paper, generalized higher-order contingent (adjacent) derivatives of set-valued maps are introduced and some of their properties are discussed. Under no any convexity assumptions, necessary and sufficient optimality conditions are obtained for weakly efficient solutions of set-valued optimization problems by employing the generalized higher-order derivatives.