Necessary conditions for approximate solutions of vector and set optimization problems with variable domination structure

Vector optimization with variable domination structures is a growing up and expanding field of applied mathematics that deals with optimization problems where the domination structure is given by a set-valued map acting between abstract or finite-dimensional spaces. Interesting and important applications of vector optimization with variable domination structure arise in economics, psychology, capability of human behavior, in portfolio management, location theory and radiotherapy treatment in medicine. We give a detailed discussion of solution concepts for problems with variable domination structures based on the (pre-) domination structures. We present certain modifications of translation invariant functionals and show characterizations of solutions to vector optimization problems with variable domination structure by means of translation invariant functionals as well as corresponding modifications. Furthermore, we introduce several concepts for approximate solutions to vector optimization problems with variable domination structures and show corresponding characterizations by means of translation invariant functionals. These results are very useful for further research in the field of vector optimization with variable domination structure, especially for deriving optimality conditions, duality assertions and numerical procedures.

In this chapter, vector optimization problems in linear spaces are studied. Here the notion of vector optimization is used in a very general way. A vector optimization problem with a variable domination structure is equivalent to the problem of finding nondominated solutions in a set related to some arbitrary binary relation. Each decision problem can be shown to be of this type. We introduce the basic concept for variable domination structures and connect it to decision problems. Optimal decisions are investigated as optima w.r.t. (with regard to) the preference relation, and the results can straightforwardly be transferred to optima w.r.t. arbitrary binary relations. It will be proved that—even for variable domination structures—efficient points can be helpful in finding the best solutions. Later on, we investigate the case where the domination structure can be described by a single set, namely the domination set of the vector optimization problem. We define efficiency w.r.t. reference sets which are not necessarily domination sets and give a thorough motivation for this approach. Properties of such efficient point sets are investigated. We study efficient and weakly efficient solutions of vector optimization problems, including surrogates for the weakly efficient point set and problems with uncertainties or perturbations. Scalarization results are proved, with an emphasis on translation invariant functions and implications for norms. Beside this, some basic properties of efficient and weakly efficient solutions are examined, especially their existence. The results imply statements for properly efficient solutions and related subsets of the efficient point set.

Many applications require the optimization of multiple conflicting goals at the same time. Such a problem can be modeled as a vector optimization problem. Vector optimization deals with the problem of finding efficient elements of a vector-valued function. In that sense, vector optimization generalizes the concept of scalar optimization. In scalar optimization, there is only one concept for efficiency which characterizes efficient elements, namely the solution which generates the smallest function value. But, due to the lack of a total order in general spaces, order relations that are defined within the optimality concept need to be chosen. In this chapter, we discuss several solution concepts for a vector optimization problem. In particular, solution concepts for vector optimization problem equipped with a variable domination structure are studied. Moreover, we present some existence results for solutions of vector optimization problems.

The aim of this paper is applying the scalarization technique to study some properties of the vector optimization problems under variable domination structure. We first introduce a nonlinear scalarization function of the vector-valued map and then study the relationships between the vector optimization problems under variable domination structure and its scalarized optimization problems. Moreover, we give the notions of DH-well-posedness and B-well-posedness under variable domination structure and prove that there exists a class of scalar problems whose well-posedness properties are equivalent to that of the original vector optimization problem.

Background: Nowadays, variable domination structure is instrumental in studying multiobjective decision making problems. We investigate multiobjective location problems with respect to variable domination structure and its applications in supply chain management. Methods: We formulate practical problems in supply chain management as an optimization problem with a variable domination structure. Moreover, we present the mathematical methods to solve such problems. We investigate two kinds of solutions derived from the concept of minimal and nondominated solutions from vector optimization problems with respect to variable domination structure. Furthermore, we explore how these solution concepts are characterized in practical problems. Results: We ex- pose how those solutions are beneficial in practical problems. However, these results hold true for multiobjective decision making problems with a continuous feasible set; we present a practical problem in the case of a finite set of feasible locations. Conclusions: In many multiobjective location problems, each location’s characterizations, preferences, and restrictions are involved in the decision making process. This study investigates the decision making problems, where different preferences of objective functions at each location are assumed. Moreover, we present a numerical experiment for selecting a new hub airport.

In this paper, we investigate a unified method to characterize robustness for uncertain vector optimization problems with variable domination structures in the framework of linear topological image spaces. We introduce new concepts of robustness for uncertain vector optimization problems with variable domination structure where the domination structure is given by a general set-valued map. By exploiting linear and nonlinear scalarization approaches for convex and nonconvex domination structures, respectively, as well as image space analysis, a series of robust optimality conditions in the context of variable domination structures are obtained under weak assumptions concerning the domination structure. Furthermore, some examples are provided to demonstrate the validity of the main results obtained. Some of the results in this paper generalize the corresponding ones in recent literature.

Solving set optimization problems based on the concept of null set

For generating solutions to vector optimization problems via algorithms based on a scalarization, the monotonicity properties of the scalarizing functionals are important. In this chapter, we present Benson’s Outer Approximation Algorithm that uses a scalarization by means of translation invariant functionals. Furthermore, we present proximal-point algorithms as well as an adaptive algorithm for solving vector optimization problems where translation invariant functionals are involved. We show that a scalarization by means of translation invariant functionals is useful for deriving an algorithm for solving set-valued optimization problems. Finally, we derive algorithms for solving vector optimization problems with variable domination structure using an extension of translation invariant functionals.

We study multi-objective location problems with respect to a variable domination structure. To solve such problems, we apply minimal and nondominated solutions from vector optimization problems with respect to a variable domination structure. We investigate these solution concepts to be used in multi-objective location problems.These solution concepts are proposed for multi-objective location problems where variable preferences of objective functions at each location are assumed; our results are applied for selecting a location to establish a new branch.

In this paper, we study the optimality conditions for set optimization problems with set criterion. Firstly, we establish a few important properties of the Minkowski difference for sets. Then, we introduce the generalized second-order lower radial epiderivative for a set-valued maps by Minkowski difference, and discuss some of its properties. Finally, by virtue of the generalized second-order lower radial epiderivatives and the generalized second-order radial epiderivatives, we establish the necessary optimality conditions and sufficient optimality conditions of approximate Benson proper efficient solutions and approximate weakly minimal solutions of unconstrained set optimization problems without convexity conditions, respectively. Some examples are provided to illustrate the main results obtained.

The aim of this paper is to study stability of the sets of l-minimal approximate solutions and weak l-minimal approximate solutions for set optimization problems with respect to the perturbations of feasible sets and objective mappings. We introduce a new metric between two set-valued mappings by utilizing a Hausdorff-type distance proposed by Han [A Hausdorff-type distance, the Clarke generalized directional derivative and applications in set optimization problems. Appl Anal. 2022;101:1243–1260]. The new metric between two set-valued mappings allows us to discuss set optimization problems with respect to the perturbation of objective mappings. Then, we establish semicontinuity and Lipschitz continuity of l-minimal approximate solution mapping and weak l-minimal approximate solution mapping to parametric set optimization problems by using the scalarization method and a density result. Finally, our main results are applied to stability of the approximate solution sets for vector optimization problems.

In this article, we propose a quasi-Newton method for unconstrained set optimization problems to find its weakly minimal solutions with respect to lower set-less ordering. The set-valued objective mapping under consideration is given by a finite number of vector-valued functions that are twice continuously differentiable. At first, we derive a necessary optimality condition for identifying weakly minimal solutions for the considered set optimization problem with the help of a family of vector optimization problems and the Gerstewitz scalarizing function. To find the necessary optimality condition for weak minimal points with the help of the proposed quasi-Newton method, we use the concept of partition and formulate a family of vector optimization problems. The evaluation of necessary optimality condition for finding the weakly minimal points involves the computation of the approximate Hessian of every objective function, which is done by a quasi-Newton scheme for vector optimization problems. In the proposed quasi-Newton method, we derive a sequence of iterative points that exhibits convergence to a point which satisfies the derived necessary optimality condition for weakly minimal points. After that, we find a descent direction for a suitably chosen vector optimization problem from this family of vector optimization problems and update from the current iterate to the next iterate. The proposed quasi-Newton method for set optimization problems is not a direct extension of that for vector optimization problems, as the selected vector optimization problem varies across the iterates. The well-definedness and convergence of the proposed method are analyzed. The convergence of the proposed algorithm under some regularity condition of the stationary points, a condition on nonstationary points, the boundedness of the norm of quasi-Newton direction, and the existence of step length that satisfies the Armijo condition are derived. We obtain a local superlinear convergence of the proposed method under uniform continuity of the Hessian approximation function. Lastly, some numerical examples are given to exhibit the performance of the proposed method. Also, the performance of the proposed method with the existing steepest-descent method is given.

In this paper, some properties of the lower (upper) semi-continuity for set-valued maps taking values in an abstract pre-ordered set are showed, which are then applied to study the existence of solutions for abstract set optimization problems. Moreover, some relationships between minimal solutions for abstract set-valued optimization problems with the vector and set criteria are given under mild conditions. As applications, existence results of solutions for set optimization problems are applied to obtain the existence of saddle points for the set-valued map taking values in the pre-ordered set with the set criterion and of solutions for vector optimization problems whose image space is a real vector space not necessarily endowed with a topology.

In this paper, approximate solutions of vector optimization problems are analyzed via a metrically consistent ?-efficient concept. Several properties of the ?-efficient set are studied. By scalarization, necessary and sufficient conditions for approximate solutions of convex and nonconvex vector optimization problems are provided; a characterization is obtained via generalized Chebyshev norms, attaining the same precision in the vector problem as in the scalarization.

In the present paper, we consider fuzzy optimization problems which involve fuzzy sets only in the objective mappings, and give two concepts of optimal solutions which are non-dominated solutions and weak non-dominated solutions based on orderings of fuzzy sets. First, by using level sets of fuzzy sets, the fuzzy optimization problems treated in this paper are reduced to set optimization problems, and relationships between (weak) non-dominated solutions of the fuzzy optimization problems and the reduced set optimization problems are derived. Next, the set optimization problems are reduced to scalar optimization problems which can be regarded as scalarization of the fuzzy optimization problems. Then, relationships between non-dominated solutions of the fuzzy optimization problems and optimal solutions of the reduced scalar optimization problems are derived.