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The multi-objective integer programming problem often occurs in multi-criteria decision-making situations, where the decision variables are integers. In the present paper, we have discussed an algorithm for finding all efficient solutions of a multi-objective integer quadratic programming problem. The proposed algorithm is based on the aspect that efficient solutions of a multi-objective integer quadratic programming problem can be obtained by enumerating ranked solutions of an integer quadratic programming problem. For determining ranked solutions of an integer quadratic programming problem, we have constructed a related integer linear programming problem and from ranked solutions of this integer linear programming problem, ranked solutions of the original integer quadratic programming problem are generated. Theoretically, we have shown that the developed method generates the set of all efficient solutions in a finite number of steps, and numerically we have elaborated the working of our algorithm and compared our results with existing algorithms. Further, we have analyzed that the developed method is efficient for solving a multi-objective integer quadratic programming problem with a large number of constraints, variables and objectives. [ABSTRACT FROM AUTHOR]

In this paper we consider multi-objective optimization problems over a box. Several computational approaches to solve these problems have been proposed in the literature, that broadly fall into two main classes: evolutionary methods, which are usually very good at exploring the feasible region and retrieving good solutions even in the nonconvex case, and descent methods, which excel in efficiently approximating good quality solutions. In this paper, first we confirm, through numerical experiments, the advantages and disadvantages of these approaches. Then we propose a new method which combines the good features of both. The resulting algorithm, which we call Non-dominated Sorting Memetic Algorithm, besides enjoying interesting theoretical properties, excels in all of the numerical tests we performed on several, widely employed, test functions. [ABSTRACT FROM AUTHOR]

In this paper, we perform optimality conditions and sensitivity analysis for parametric nonconvex minimax programming problems. Our aim is to study the necessary optimality conditions by using the Mordukhovich (limiting) subdifferential and to give upper estimations for the Mordukhovich subdifferential of the optimal value function in the problem under consideration. The optimality conditions and sensitivity analysis are obtained by using upper estimates for Mordukhovich subdifferentials of the maximum function. The results on optimality conditions are then applied to parametric multiobjective optimization problems. An example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, a class of directionally differentiable multiobjective programming problems with inequality, equality and vanishing constraints is considered. Under both the Abadie constraint qualification and the modified Abadie constraint qualification, the Karush–Kuhn–Tucker type necessary optimality conditions are established for such nondifferentiable vector optimization problems by using the nonlinear version Gordan theorem of the alternative for convex functions. Further, the sufficient optimality conditions for such directionally differentiable multiobjective programming problems with vanishing constraints are proved under convexity hypotheses. Furthermore, vector Wolfe dual problem is defined for the considered directionally differentiable multiobjective programming problem vanishing constraints and several duality theorems are established also under appropriate convexity hypotheses. [ABSTRACT FROM AUTHOR]

In this paper, a new method for computing an enclosure of the nondominated set of multiobjective mixed-integer quadratically constrained programs without any convexity requirements is presented. In fact, our criterion space method makes use of piecewise linear relaxations in order to bypass the nonconvexity of the original problem. The method chooses adaptively which level of relaxation is needed in which parts of the image space. Furthermore, it is guaranteed that after finitely many iterations, an enclosure of the nondominated set of prescribed quality is returned. We demonstrate the advantages of this approach by applying it to multiobjective energy supply network problems. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new scheme for approximating the weakly efficient solution set for a class of vector optimization problems with rational objectives over a feasible set defined by finitely many polynomial inequalities. More precisely, we present a procedure to obtain a sequence of explicit approximations of the weakly efficient solution set of the problem in question. Each approximation is the intersection of the sublevel set of a single polynomial and the feasible set. To this end, we make use of the achievement function associated with the considered problem and construct polynomial approximations of it over the feasible set from above. Remarkably, the construction can be converted to semidefinite programming problems. Several nontrivial examples are designed to illustrate the proposed new scheme. [ABSTRACT FROM AUTHOR]

An appropriate and sustainable waste management plan is required in different scenarios for global development. The main goal of this paper is to evaluate a waste management problem by investigating an integrated multi-objective environment through solid transportation problem. To develop sustainability, three objective functions are optimized by choosing as cost for economical opportunity, time for social satisfaction and carbon emission for environmental view. Cap and trade policy is regarded here to minimize carbon emission and to provide some economical opportunities to the system. To control hesitancy of this scenario, twofold uncertainty (type-2 intuitionistic fuzzy) is incorporated here, and this uncertainty is defuzzified by a ranking operator. A strategy is proposed here to optimize three factors of sustainability by an intellectual model formulation of solid waste management. To check the appropriateness of the proposed model, two numerical problems are evaluated using two advanced methods, namely, neutrosophic linear programming and ϵ -constraint method. The Pareto-optimal solutions are derived by the variation of cap value and fulfilling the criteria of sustainability. The obtained results indicate that cap and trade policy or waste management, or both are highly sophisticated for applying in real-world application. The overall conclusions recommend that a government or NGO should encourage transportation system, or the industry to minimize carbon emission by utilizing several carbon policies. It can assist to establish different new project of waste management in a discrete environment, based on sustainability. [ABSTRACT FROM AUTHOR]

In data envelopment analysis, methods for constructing sections of the frontier have been recently proposed to visualize the production possibility set. The aim of this paper is to develop, prove and test the methods for the visualization of production possibility sets using parallel computations. In this paper, a general scheme of the algorithms for constructing sections (visualization) of production possibility set is proposed. In fact, the algorithm breaks the original large-scale problems into parallel threads, working independently, then the piecewise solution is combined into a global solution. An algorithm for constructing a generalized production function is described in detail. [ABSTRACT FROM AUTHOR]

In this paper, for a robust nonsmooth semi-infinite objective optimization problem associated with data uncertainty, some constraint qualifications (CQs): Abadie CQ, Mangasarian-Fromovitz CQ, and Pshenichnyi-Levin-Valadire CQ are proposed. Sufficient conditions for them are also derived. Under these CQs, we establish both necessary and sufficient conditions for robust weak Pareto, Pareto, and Benson proper solutions. These conditions are the forms of Karush-Kuhn-Tucker rule. Moreover, the Wolfe and Mond-Weir duality schemes are also addressed. Finally, we employ the obtained results to present some conditions for linear programming. Examples are provided for analyzing and illustrating our results. [ABSTRACT FROM AUTHOR]

In this paper, the behavior of the solutions of a multi-objective optimization problem, whose the objective functions are perturbed by adding a small linear term, is analyzed. In this regard, a new notion of Lipschitzian stability, by means of the Aubin property of the solution set, is defined. Lipschitz stable locally efficient solutions, as generalization of tilt/full stable solutions, are introduced and characterized by modern variational analysis tools. Applying the weighted sum method, the relationships between these solutions and full-stable local optimal solutions of the scalarized problem are investigated. The key tools in deriving our results come from the first- and second-order variational analysis. [ABSTRACT FROM AUTHOR]

In this paper, we derive second-order necessary optimality conditions and second-order sufficient optimality condition for locally Pareto solutions to a class of multi-objective optimal control problems with mixed pointwise constraints. The necessary optimality conditions for the multi-objective optimal control problem are established via optimality conditions of a specific vector optimization problem in Banach spaces, which is proved directly without using the scalarization method. [ABSTRACT FROM AUTHOR]

In this paper, we study an uncertain inequality system, where the input data are uncertain and belong to prescribed uncertainty sets. Using the deterministic approach in robust optimization, we treat this uncertain system by examining the so-called robust system. This approach enables us to compute the second order tangent sets for the solution set of the robust system and then obtain the second order epi-subderivative for the indicator function of its solution set. In this way, we are able to calculate the graphical derivative for the normal cone mapping of solution set of the robust system under certain qualification conditions. As applications, we establish second order necessary and sufficient optimality conditions, and derive necessary and sufficient conditions for stability properties such as the isolated calmness of optimization problems involving uncertain constraints under weak qualification conditions. [ABSTRACT FROM AUTHOR]

The search region in multiobjective optimization problems is a part of the objective space where nondominated points could lie. It plays an important role in the generation of the nondominated set of multiobjective combinatorial optimization (MOCO) problems. In this paper, we establish the representation of the search region by half-open polyblocks (a variant concept of "polyblock" in monotonic optimization) and propose a new procedure for updating the search region. We also study the impact of stack policies to the new procedure and the existing methods that update the search region. Stack policies are then analyzed, pointing out their performance effectiveness by means of the results of rich computational experiments on finding the whole set of nondominated points of MOCO problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study optimality conditions for strict local minima to constrained mathematical problems governed by scalar and vectorial mappings. Unlike other papers in literature dealing with strict efficiency, we work here with mappings defined on infinite dimensional normed vector spaces. Firstly, we (mainly) consider the case of nonsmooth scalar mappings and we use the Fréchet and Mordukhovich subdifferentials in order to provide optimality conditions. Secondly, we present some methods to reduce the study of strict vectorial minima to the case of strict scalar minima by means of some scalarization techniques. In this vectorial framework we treat separately the case where the ordering cone has non-empty interior and the case where it has empty interior. [ABSTRACT FROM AUTHOR]

This work is an attempt to develop multiobjective versions of some well-known single objective quasi-Newton methods, including BFGS, self-scaling BFGS (SS-BFGS), and the Huang BFGS (H-BFGS). A comprehensive and comparative study of these methods is presented in this paper. The Armijo line search is used for the implementation of these methods. The numerical results show that the Armijo rule does not work the same way for the multiobjective case as for the single objective case, because, in this case, it imposes a large computational effort and significantly decreases the speed of convergence in contrast to the single objective case. Hence, we consider two cases of all multi-objective versions of quasi-Newton methods: in the presence of the Armijo line search and in the absence of any line search. Moreover, the convergence of these methods without using any line search under some mild conditions is shown. Also, by introducing a multiobjective subproblem for finding the quasi-Newton multiobjective search direction, a simple representation of the Karush-Kuhn-Tucker conditions is derived. The H-BFGS quasi-Newton multiobjective optimization method provides a higher-order accuracy in approximating the second order curvature of the problem functions than the BFGS and SS-BFGS methods. Thus, this method has some benefits compared to the other methods as shown in the numerical results. All mentioned methods proposed in this paper are evaluated and compared with each other in different aspects. To do so, some well-known test problems and performance assessment criteria are employed. Moreover, these methods are compared with each other with regard to the expended CPU time, the number of iterations, and the number of function evaluations. [ABSTRACT FROM AUTHOR]

In this paper we introduce a definition of approximate Pareto efficient solution as well as a necessary condition for such solutions in the multiobjective setting on Riemannian manifolds. We also propose an inexact proximal point method for nonsmooth multiobjective optimization in the Riemannian context by using the notion of approximate solution. The main convergence result ensures that each cluster point (if any) of any sequence generated by the method is a Pareto critical point. Furthermore, when the problem is convex on a Hadamard manifold, full convergence of the method for a weak Pareto efficient solution is obtained. As an application, we show how a Pareto critical point can be reached as a limit of traps in the context of the variational rationality approach of stay and change human dynamics. [ABSTRACT FROM AUTHOR]

In this paper, the class of differentiable semi-infinite multiobjective programming problems with vanishing constraints is considered. Both Karush–Kuhn–Tucker necessary optimality conditions and, under appropriate invexity hypotheses, sufficient optimality conditions are proved for such nonconvex smooth vector optimization problems. Further, vector duals in the sense of Mond–Weir are defined for the considered differentiable semi-infinite multiobjective programming problems with vanishing constraints and several duality results are established also under invexity hypotheses. [ABSTRACT FROM AUTHOR]

In this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations. [ABSTRACT FROM AUTHOR]

This paper is concerned with a multiobjective convex polynomial problem, where the objective and constraint functions are first-order scaled diagonally dominant sums-of-squares convex polynomials. We first establish necessary and sufficient optimality criteria in terms of second-order cone (SOC) conditions for (weak) efficiencies of the underlying multiobjective optimization problem. We then show that the obtained result provides us a way to find (weak) efficient solutions of the multiobjective program by solving a scalar second-order cone programming relaxation problem of a given weighted-sum optimization problem. In addition, we propose a dual multiobjective problem by means of SOC conditions to the multiobjective optimization problem and examine weak, strong and converse duality relations. [ABSTRACT FROM AUTHOR]

This paper establishes an evaluation index system for the effectiveness of online teaching of ideological and political theory courses in colleges and universities. Then the AHP method is used to study the weight of each evaluation index. The review found that the online education of ideological and political courses in universities is generally “strong” under the current situation. Based on this, this paper comprehensively evaluates college students’ ideological and political work. Educational institutions can then develop targeted improvement plans on a quantitative basis. The results of this paper lay a foundation for the implementation of precision teaching.

Improperly efficient solutions in the sense of Geoffrion in linear fractional vector optimization problems with unbounded constraint sets are studied systematically for the first time in this paper. We give two sets of conditions which assure that all the efficient solutions of a given problem are improperly efficient. We also obtain necessary conditions for an efficient solution to be improperly efficient. As a result, we have new sufficient conditions for Geoffrion's proper efficiency. The obtained results enrich our knowledge on properly efficient solutions in linear fractional vector optimization. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of Painlevé−Kuratowski convergence of constraint sets, minimal and efficient solution sets of a unified semi-infinite vector optimization problem. The convergence is established under the functional perturbations of objective function and constraint maps assuming the continuous convergence of the objective functions and uniform gamma convergence of the constraint maps. Further, examples are formulated to analyze the significance of assumptions in the main results. [ABSTRACT FROM AUTHOR]