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In this paper, we first introduce the notion of (incident) higher-order generalized epiderivative for a set-valued map and obtain a crucial result of the epiderivative. Then we use this result to establish the higher-order sufficient and necessary optimality conditions of a constrained set-valued optimization problem. By virtue of the epiderivatives and the optimality conditions, we establish the higher-order mixed duality problem for the set-valued optimization problem and obtain the corresponding duality theorems. [ABSTRACT FROM AUTHOR]

This paper deals with a kind of nonconvex optimistic bilevel optimization programs. In some process of dealing this kind of bilevel programs, difficulties are essentially moved to estimating for coderivative of the solution map. To deal with these difficulties, we use value function of the lower level problem and its modifications as implicit functions to describe the solution map. By applying techniques in variational analysis, we give estimates for coderivative of the solution map. Then, we will show the applications in optimality conditions for these bilevel programs which we derived by using the extremal principle. [ABSTRACT FROM AUTHOR]

Set-valued optimization using the set approach is a research topic of high interest due to its practical relevance and numerous interdependencies to other fields of optimization. However, it is a very difficult task to solve these optimization problems even for specific cases. In this paper, we study set-valued optimization problems and develop a multiobjective optimization problem that is strongly related to it. We prove that the set of weakly minimal solutions of this subproblem is closely related to the set of weakly minimal elements of the set-valued optimization problem and that these sets can get arbitrarily close in a certain sense. Subsequently, we introduce a concept of approximations of the solution set of the set-valued optimization problem. We define a quality measure in the image space that can be used to compare finite approximations of this kind and outline a procedure to enhance a given approximation. We conclude the paper with some numerical examples. [ABSTRACT FROM AUTHOR]

Well-posedness for optimization problems is a well-known notion and has been studied extensively for scalar, vector and set-valued optimization problems. There is a broad classification in terms of pointwise and global well-posedness notions in vector and set-valued optimization problems. We have focused on global well-posedness for set-valued optimization problems in this paper. A number of notions of global well-posedness for set-valued optimization problems already exist in the literature. However, we found equivalence between some existing notions of global well-posedness for set-valued optimization problems and also found scope of improving and extending the research in that field. That has been the first aim of this paper. On the other hand, robust approach towards uncertain optimization problems is another growing area of research. The well-posedness for the robust counterparts have been explored in very few papers, and that too only in the scalar and vector cases (see (Anh et al. in Ann Oper Res 295(2):517–533, 2020), (Crespi et al. in Ann Oper Res 251(1–2):89–104, 2017)). Therefore, the second aim of this paper is to study some global well-posedness properties of the robust formulation of uncertain set-valued optimization problems that generalize the concept of the well-posedness of robust formulation of uncertain vector optimization problems as discussed in Anh et al. (Ann Oper Res 295(2):517–533, 2020), Crespi et al. (Ann Oper Res 251(1–2):89–104, 2017). [ABSTRACT FROM AUTHOR]

The set criterion is an appropriate defining approach regarding the solutions for the set-valued optimization problems. By using approximations as generalized derivatives of set-valued mappings, we establish necessary optimality conditions for a constrained set-valued optimization problem in the sense of set optimization in terms of asymptotical pointwise compact approximations. Sufficient optimality conditions are then obtained through first-order strong approximations of data set-valued mappings. [ABSTRACT FROM AUTHOR]

The usual approximation solutions with variable ordering structures fail in some cases such as Examples 2.1 and 2.2, where the intersection of all ordering cones equals to the zero element. Inspired by this, we introduce a new approximate nondominated solution depending on a variable point substituting for a fixed point, which generalize the usual solutions. By proposing two types of separation bi-functionals, optimality characterizations in a unified way are concluded for various approximate nondominated solutions. Augmented dual cones and max scalarizing functional are proved to associate closely with some specific separation bi-functionals. Considering only weakly nondominated or nondominated minimizers, we also derive the Fermat rules based on the tools of variational analysis. Particularly, the important setting for variable ordering structures equipped with Bishop-Phelps cones are examined based on our theories. [ABSTRACT FROM AUTHOR]

We present higher order necessary conditions for a model of welfare economics, where the preference mapping has a star-shape property. We assume that the preferences can be satiable and can be described by an arbitrary preference set, without the use of utility functions. These conditions are formulated in terms of higher-order directional derivatives of multivalued mappings, and the variable domination structure is not given by cones. [ABSTRACT FROM AUTHOR]

In this work we adapt the main results from Khanh and Tung [First and second-order optimality conditions without differentiability in multivalued vector optimization. Positivity. 2015;19:817–841] for a general set-valued optimization problem to an optimistic bilevel set-valued programming problem as an optimization problem with implicitly given set-valued constraints. Using the optimal value function, we convert our problem into a one level set-valued optimization problem with general inequality constraints and derive both necessary conditions and sufficient conditions of order one and two. The main tools we exploit are approximations of set-valued mappings. To illustrate the obtained results, some examples are given. [ABSTRACT FROM AUTHOR]

In this paper we extend the notion of stochastic efficiency and inefficiency in portfolio optimization to the case of incomplete information by means of set-valued probabilities. The notion of set-valued probability models the concept of incomplete information about the underlying probability space and the probability associated with each scenario. Unlike other approaches in literature, our notion of inefficiency is introduced by means of the Monge–Kantorovich metric. We provide some numerical examples to illustrate this approach. [ABSTRACT FROM AUTHOR]

In this paper, we give a comparison among some notions of weak sharp minima introduced in Amahroq et al. [Le matematiche J.73 (2018) 99–114], Durea and Strugariu [Nonlinear Anal.73 (2010) 2148–2157] and Zhu et al. [Set-Valued Var. Anal.20 (2012) 637–666] for set-valued optimization problems. Besides, we establish sharp Lagrange multiplier rules for general constrained set-valued optimization problems involving new scalarization functionals based on the oriented distance function. Moreover, we provide sufficient optimality conditions for the considered problems without any convexity assumptions. [ABSTRACT FROM AUTHOR]

Well-posedness for optimization problems is a well-known notion and has been studied extensively for scalar, vector, and set-valued optimization problems. For the set-valued case, there are many subdivisions: firstly in terms of pointwise notion and global notion and secondly in terms of the solution concepts, like the vector approach, the set-relation approach, etc. Various definitions of pointwise well-posedness for a set-valued optimization problem in the set-relation approach have been proposed in the literature. Here we do a comparative study and suggest modifications in some existing results. We also introduce a new pointwise well-posedness and discuss its properties and connection with others. [ABSTRACT FROM AUTHOR]

In this paper, we propose new concepts of sharp minimality in set-valued optimization problems by means of the pseudo-relative interior, namely pseudo-relative ϕ -sharp minimizers. Based on this notion of minimality, we extend the existence result of a unique minimum of uniformly convex real-valued functions proved by Zălinescu in [25] to vector-valued as well as set-valued maps. Additionally, we provide some existence results for weak sharp minimizers in the sense of Durea and Strugariu [9]. [ABSTRACT FROM AUTHOR]

In the recent past, finding robust solutions for optimization problems contaminated with uncertainties has been topical and has been investigated in the literature for scalar and multi-objective/vector-valued optimization problems. In this paper, we introduce various types of robustness concept for set-valued optimization, such as min–max set robustness, optimistic set robustness, highly set robustness, flimsily set robustness, multi-scenario set robustness. We study some existence results for corresponding concepts of solution and establish some relationship among them. [ABSTRACT FROM AUTHOR]

In this study, set-valued optimization problems are considered with respect to m 1 -order relation, which is a partial order relation on the family of nonempty and bounded sets. Four subdifferentials are defined for set-valued optimization maps. Some existence theorems and relations among these subdifferentials are obtained. By using them, necessary and sufficient optimality conditions are presented for set-valued optimization problems with respect to m 1 -order relation. [ABSTRACT FROM AUTHOR]

Abstract In this paper, firstly, a new property of the cone subpreinvex set-valued map involving the generalized contingent epiderivative is obtained. As an application of this property, a sufficient optimality condition for constrained set-valued optimization problem in the sense of globally proper efficiency is derived. Finally, we establish the relations between the globally proper efficiency of the set-valued optimization problem and the globally proper efficiency of the vector variational inequality.

This paper deals with optimality conditions and duality theory for vector optimization involving non-convex set-valued maps. Firstly, under the assumption of nearly cone-subconvexlike property for set-valued maps, the necessary and sufficient optimality conditions in terms of limit sets are derived for local weak minimizers of a set-valued constraint optimization problem. Then, applications to Mond-Weir type and Wolfe type dual problems are presented.

A new kind of tangent derivative, M-derivative, for set-valued function is introduced with help of a modified Dubovitskij-Miljutin cone. Several generalized pseudoconvex set-valued functions are introduced. When both the objective function and constraint function are M-derivative, under the assumption of near conesubconvexlikeness, by applying properties of the set of strictly efficient points and a separation theorem for convex sets, Fritz John and Kuhn-Tucker necessary optimality conditions are obtained for a point pair to be a strictly efficient element of set-valued optimization problem. Under the assumption of generalized pseudoconvexity, a Kuhn-Tucker sufficient optimality condition is obtained for a point pair to be a strictly efficient element of set-valued optimization problem. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to establish higher-order Karush–Kuhn–Tucker higher-order necessary optimality conditions for set-valued optimization where the derivatives of objective and constraint functions are separated. We first introduce concepts of higher-order weak lower inner epiderivatives for set-valued maps and discuss some useful properties about new epiderivatives, for instance, convexity, subadditivity and chain rule. With the help of the new concept and its properties, we establish higher-order Karush–Kuhn–Tucker necessary optimality conditions which is the classical type Karush–Kuhn–Tucker optimality conditions and improve and enhance some recent existing results in the literatures. Several examples are provided to illustrate our results. Finally, we provide weak and strong duality theorems in set-valued optimization. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of efficient elements for set-valued maps. We propose two new notions of relative weak ϵ -efficient element and strict relative weak ϵ -efficient element of set-valued maps and provide new necessary optimality conditions for the proposed concepts. We provide existence results for efficient elements. The critical ingredients for the existence results for efficient elements are the well-known separation arguments and Fan's lemma. As an application of the existence results, we derive relationships between the efficiency concepts and the local optimizers of certain optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a new scalar representation of set-valued optimization problems by using a criterion of solution introduced by Kuroiwa, this criterion is a weaker version of Pareto efficient solution. Following this scalarization, we establish necessary optimality conditions in terms of Fritz John multipliers as well as Karush-Kuhn-Tucker multipliers. Moreover, we give sufficient optimality conditions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the paper, we develop sum and chain rules of the generalized contingent derivative for set-valued mappings. Then, their applications to sensitivity analysis and optimality conditions for some particular optimization problems are given. Our results extend some recent existing ones in the literature. [ABSTRACT FROM AUTHOR]

In this work, set-valued optimization problems are considered according to an order relation, which is a partial order on the family such that contains nonempty bounded sets of the space. A generalized convexity is defined for set-valued mapping by using the partial order relation. Nonsmooth variational inequality problems are introduced with the aid of M-directionally derivative. Some optimality criteria including the necessary and sufficient optimality conditions are obtained for mentioned optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we provide the augmented weighted Tschebyscheff scalarization to computing robust efficient solutions for uncertain multiobjective optimization problems (UMOPs). Furthermore, we propose two generalized robustness concepts called the minmax certainly nondominated ordered robustness and the certainly less ordered robustness to general spaces and cones in the context of set orderings. Some explanations for both concepts are given, as well as the relations between them and several existing robustness are clearly described. Finally, the corresponding relationships to set-valued optimization are well revealed. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the nonemptiness of the set of weak minimal elements for a nonempty subset of a linear space. Moreover, we obtain some existence results for a nonconvex set-valued optimization problem under weaker topological conditions. [ABSTRACT FROM AUTHOR]

Optimality conditions are established in terms of Lagrange–Fritz–John multipliers as well as Lagrange–Kuhn–Tucker multipliers for set optimization problems (without any convexity assumption) by using new scalarization techniques. Additionally, we indicate how these results may be applied to some particular weak vector equilibrium problems. [ABSTRACT FROM AUTHOR]

We introduce and study a notion of directional Pareto minimality with respect to a set that generalizes the classical concept of Pareto efficiency. Then, we give separate necessary and sufficient conditions for the newly introduced efficiency and several situations, concerning the objective mapping and the constraints, are considered. In order to investigate different cases, we adapt some well-known constructions of generalized differentiation; the connections with some recent directional regularities come naturally into play. As a consequence, several techniques from the study of genuine Pareto minima are considered in our specific situation. [ABSTRACT FROM AUTHOR]

In this paper we extend the notion of minimal solutions for a vector optimization problem considered by Flores-Bazán et al. (J Optim Theory Appl 164:455–478, 2015) to a set-valued optimization problem, with both vector and set solution criteria. Also, we extend the Gerstewitz function proposed by Hernández and Rodríguez-Marín (J Math Anal Appl 325:1–18, 2007) and use it to scalarize minimal solutions with respect to set criterion. We also provide an existence result of minimal solutions with set criterion. Finally, we investigate links between the minimal solutions with respect to vector criterion and set criterion. [ABSTRACT FROM AUTHOR]

In this paper, we supply the power set P(Z) of a partially ordered normed space Z with a transitive and irreflexive binary relation which allows us to introduce a notion of open intervals on P(Z) from which we construct a topology on the set of lower bounded subsets of Z. From this topology, we derive a concept of set convergence that is compatible with the strict ordering on P(Z) and, taking advantage of its properties, we prove several stability results for minimal sets and minimal solutions to set-valued optimization problems. [ABSTRACT FROM AUTHOR]

Abstract We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively. Some properties of second-order tangent derivatives are discussed, using which second-order necessary optimality conditions are established for a point pair to be a Henig efficient element of a set-valued optimization problem, and in the expressions the second-order tangent derivatives of the objective function and the constraint function are separated.

In this paper we introduce a notion of minimal solutions for set-valued optimization problem in terms of improvement sets, by unifying a solution notion, introduced by Kuroiwa [15] for set-valued problems, and a notion of optimal solutions in terms of improvement sets, introduced by Chicco et al. [4] for vector optimization problems. We provide existence theorems for these solutions, and establish lower convergence of the minimal solution sets in the sense of Painlevé-Kuratowski.

The aim of this work is twofold. First, we establish sum rules for the directionally coderivatives of multifunctions and intersection rules for the directionally limiting normal cones. Then, we apply the provided formulas to derive directionally necessary conditions for a set-valued optimization problem with equilibrium constraints. [ABSTRACT FROM AUTHOR]

We consider a scalar stochastic linear optimization problem subject to linear constraints. We introduce the notion of deterministic equivalent formulation when the underlying probability space is equipped with a probability multimeasure. The initial problem is then transformed into a set-valued optimization problem with linear constraints. We also provide a method for estimating the expected value with respect to a probability multimeasure and prove extensions of the classical strong law of large numbers, the Glivenko-Cantelli theorem, and the central limit theorem to this setting. The notion of sampling with respect to a probability multimeasure and the definition of cumulative distribution multifunction are also discussed. Finally, we show some properties of the deterministic equivalent problem. [ABSTRACT FROM AUTHOR]

A new kind of second-order tangent derivative, second-order M -composed tangent derivative, for a set-valued function is introduced with help of a modified Dubovitskij–Miljutin cone. By using the concept, several generalized convex set-valued functions are introduced. When both the objective function and constrained function are second-order M -composed derivable, under the assumption of nearly cone-subconvexlikeness, by applying a separation theorem for convex sets, Fritz John and Kuhn–Tucker second-order necessary optimality conditions are obtained for a point pair to be a weak minimizer of set-valued optimization problem. Under the assumption of generalized pseudoconvexity, a Kuhn–Tucker second-order sufficient optimality condition is obtained for a point pair to be a weak minimizer of set-valued optimization problem. A unified second-order necessary and sufficient optimality condition is derived in terms of second-order M -composed tangent derivatives. [ABSTRACT FROM AUTHOR]

Vectorization is a technique that replaces a set-valued optimization problem with a vector optimization problem. In this work, by using an extension of the Gerstewitz function, a vectorizing function is defined to replace a given set-valued optimization problem with respect to the set less order relation. Some properties of this function are studied. Moreover, relationships between a set-valued optimization problem and a vector optimization problem, derived via vectorization of this set-valued optimization problem, are examined. Furthermore, necessary and sufficient optimality conditions are presented without any convexity assumption. [ABSTRACT FROM AUTHOR]

We introduce a class of positively homogeneous set-valued mappings, called inner prederivatives, serving as first order approximants to set-valued mappings. We prove an inverse mapping theorem involving such prederivatives and study their stability with respect to variational perturbations. Then, taking advantage of their properties we establish necessary optimality conditions for the existence of several kind of minimizers in set-valued optimization. As an application of these last results, we consider the problem of finding optimal allocations in welfare economics. Finally, to emphasize the interest of our approach, we compare the notion of inner prederivative to the related concepts of set-valued differentiation commonly used in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study different classes of generalized convex/quasiconvex set-valued maps, defined by means of the l-type and u-type preorder relations, currently used in set-valued optimization. In particular, we identify those classes of set-valued maps for which it is possible to extend the classical characterization of convex real-valued functions by quasiconvexity of their affine perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the weak, strong, and converse duality theorems for a pair of primal-dual problems in set-valued optimization concerning Q-efficient solutions. Then, duality theorems for quasi-relative efficient solutions and Henig efficient solutions are implied with Q being appropriately chosen cones. Finally, their applications to optimality conditions in the Kuhn-Tucker type are obtained. [ABSTRACT FROM AUTHOR]

In the paper, we first develop sum and chain rules of higher-order radial derivatives. By virtue of these derivatives, we establish duality theorems for a primal-dual pair in set-valued optimization. Then, their applications to optimality conditions for weakly efficient solutions are implied. Our results are more advantageous than several existing ones in the literature, especially in case of the ordering cone in the constraint space having possibly empty interior. [ABSTRACT FROM AUTHOR]

In this paper, the generalized composed radial epiderivative for set-valued maps is introduced and some of its properties are investigated. Existence conditions for the generalized composed radial epiderivative are established. [ABSTRACT FROM AUTHOR]

Certain notions of approximate weak efficient solutions are considered for a set-valued optimization problem based on vector and set criteria approaches. For approximate solutions based on the vector approach, a characterization is provided in terms of an extended Gerstewitz’s function. For the set approach case, two notions of approximate weak efficient solutions are introduced using a lower and an upper quasi order relations for sets and further compactness and stability aspects are discussed for these approximate solutions. Existence and scalarization using a generalized Gerstewitz’s function are also established for approximate solutions, based on the lower set order relation. [ABSTRACT FROM PUBLISHER]

In many real-world optimization problems, a solution cannot be realized in practice exactly as computed, e.g., it may be impossible to produce a board of exactly 3.546 mm width. Whenever computed solutions are not realized exactly but in a perturbed way, we speak of decision uncertainty. We study decision uncertainty in multiobjective optimization problems and we propose the concept of decision robust efficiency for evaluating the robustness of a solution in this case. This solution concept is defined by using the framework of set-valued maps. We prove that convexity and continuity are preserved by the resulting set-valued maps. Moreover, we obtain specific results for particular classes of objective functions that are relevant for solving the set-valued problem. We furthermore prove that decision robust efficient solutions can be found by solving a deterministic problem in case of linear objective functions. We also investigate the relationship of the proposed concept to other concepts in the literature. [ABSTRACT FROM AUTHOR]