Linear Vector Optimization Research Articles

An approach to duality via image space analysis and applications to linear vector optimization problems

In Yao et al. [Yao CL, Wang SQ, Tammer C. Construction of vector-valued weak separation functions with applications to conjugate duality in vector optimization; 2024. Submitted.], based on the idea of image space analysis and abstract convexity, we proposed a uniform framework of conjugate duality for the constrained vector optimization problems. This paper aims to apply this framework to linear vector optimizations. With the aid of a special collection of separation functions, a set-valued dual problem, which was also derived by Heyde, Löhne, Tammer in [Heyde F, Löhne A, Tammer C. The attainment of the solution of the dual program in vertices for vectorial linear programs. In: Barichard V, Ehrgott M, Gandibleux X, T'Kindt V, editors. Multiobjective programming and goal programming. 2009. p. 13–24. (Lecture Notes in Economics and Mathematical Systems; 618).; Heyde F, Löhne A, Tammer C. Set-valued duality theory for multiple objective linear programs and application to mathematical finance. Math Meth Oper Res. 2009;69:159–179.], is obtained by this approach. Within the uniform framework, we provide a new way to construct this dual model, and to investigate the duality theory of it. In this method, the strong duality is established by the separation of sets in the image space and the generalized saddle points.

Computing the recession cone of a convex upper image via convex projection

It is possible to solve unbounded convex vector optimization problems (CVOPs) in two phases: (1) computing or approximating the recession cone of the upper image and (2) solving the equivalent bounded CVOP where the ordering cone is extended based on the first phase. In this paper, we consider unbounded CVOPs and propose an alternative solution methodology to compute or approximate the recession cone of the upper image. In particular, we relate the dual of the recession cone with the Lagrange dual of weighted sum scalarization problems whenever the dual problem can be written explicitly. Computing this set requires solving a convex (or polyhedral) projection problem. We show that this methodology can be applied to semidefinite, quadratic, and linear vector optimization problems and provide some numerical examples.

Ranking of Tourism Web Sites According to Service Performance Criteria with CRITIC and MAIRCA Methods: The Case of Turkey

Multi-Criteria Decision-Making (MCDM) methods enable the systematic ranking of decision alternatives by evaluating them with more than one maximized or minimized decision criterion. Multi-Criteria Decision-Making methods provide decision-makers with forms that include specified steps for calculating the importance weights of decision criteria and ranking decision alternatives. Weighting methods for calculating criterion weights are divided into subjective and objective. CRITIC method is one of the most widely used objective weighting methods in which criterion weights are determined according to the structure of the data. The MAIRCA method, a Multi-Criteria Decision Making method based on linear vector optimization, is widely preferred in the literature, especially in recent years, as it is also suitable for criteria with zero and negative values. This study aims to rank tourism websites, which play an essential role in terms of national tourism, according to nine service performance criteria. The CRITIC method determined the importance and weight values of the criteria, and the ranking of the websites was carried out by the MAIRCA method. The most decisive decision criterion in the website ranking was the live support and number of complaints. As a result of the ranking of the websites with the MAIRCA method, tripadvisor.com and seyahatyanımda.com took the first two places, while etstur.com and setur.com took the last two.

A linearized approach for solving differentiable vector optimization problems with vanishing constraints

In this paper, two mathematical methods are used for solving a complex multicriteria optimization problem as the considered convex differentiable vector optimization problem with vanishing constraints. First of them is the linearized approach in which, for the original vector optimization problem with vanishing constraints, its associated multiobjective programming problem is constructed at the given feasible solution. Since the aforesaid multiobjective programming problem constructed in the linearized method is linear, one of the existing methods for solving linear vector optimization problems is applied for solving it. Thus, the procedure for solving the considered differentiable vector optimization problems with vanishing constraints is presented in the paper.

Proper efficiency in linear fractional vector optimization via Benson's characterization

Linear fractional vector optimization problems are special non-convex vector optimization problems. They were introduced and first studied by E.U. Choo and D.R. Atkins in the period 1982–1984. This paper investigates the properness in the sense of Geoffrion of the efficient solutions of linear fractional vector optimization problems with unbounded constraint sets. Sufficient conditions for an efficient solution to be Geoffrion's properly efficient solution are obtained via Benson's characterization [An improved definition of proper efficiency for vector maximization with respect to cones. J Math Anal Appl. 1979;71:232–241] of Geoffrion's proper efficiency.

A note on approximate proper efficiency in linear fractional vector optimization

A note on approximate proper efficiency in linear fractional vector optimization

Improperly efficient solutions in a class of vector optimization problems

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.

Fully Piecewise Linear Vector Optimization Problems

We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed spaces. Based on such a representation, we study a fully piecewise linear vector optimization problem with the objective and constraint functions being piecewise linear. To solve this problem, we divide it into some linear subproblems and structure a dimensional reduction method. Under some mild assumptions, we prove that its Pareto (resp., weak Pareto) solution set is the union of finitely many generalized polyhedra (resp., polyhedra), each of which is contained in a Pareto (resp., weak Pareto) face of some linear subproblem. Our main results are even new in the linear case and further generalize Arrow, Barankin and Blackwell’s classical results on linear vector optimization problems in the framework of finite-dimensional spaces.

Painlevé-Kuratowski convergences of the solution sets for vector optimization problems with free disposal sets

<p style='text-indent:20px;'>This paper aims to present results on the Painlev<inline-formula><tex-math id="M1">\begin{document}$ \mathrm{\acute{e}} $\end{document}</tex-math></inline-formula>-Kuratowski set-convergence of the sets of both infimal and minimal points of a sequence of perturbed vector optimization problems through free disposal sets. By assumptions of sequential compactness of the feasible sets or the uniform coerciveness of objective functions, this convergence is obtained both in the image and given spaces under the perturbations of objective functions and feasible sets. Besides, we also establish set-convergences of sequences of approximation solution sets. Applications to the stability of conic, quadratic and linear vector optimization problems are given. Some examples are provided to illustrate our results.</p>

A new approximation approach to optimality and duality for a class of nonconvex differentiable vector optimization problems

In this paper, a new approximation method for a characterization of (weak) Pareto solutions in some class of nonconvex differentiable multiobjective programming problems is introduced. In this method, an auxiliary approximated vector optimization problem is constructed at a given feasible solution of the original multiobjective programming problem. The equivalence between (weak) Pareto solutions of these two vector optimization problems is established under $$(\Phi ,\rho )$$ -invexity hypotheses. By using the introduced approximation method, it is shown in some cases that a nonlinear differentiable multiobjective programming problem can be solved by the help of some methods for solving a linear vector optimization problem. Further, the introduced approximation method is used in proving several duality results in the sense of Mond-Weir for the considered vector optimization problem.

Geoffrion’s proper efficiency in linear fractional vector optimization with unbounded constraint sets

Choo (Oper Res 32:216–220, 1984) has proved that any efficient solution of a linear fractional vector optimization problem with a bounded constraint set is properly efficient in the sense of Geoffrion. This paper studies Geoffrion’s properness of the efficient solutions of linear fractional vector optimization problems with unbounded constraint sets. By examples, we show that there exist linear fractional vector optimization problems with the efficient solution set being a proper subset of the unbounded constraint set, which have improperly efficient solutions. Then, we establish verifiable sufficient conditions for an efficient solution of a linear fractional vector optimization to be a Geoffrion properly efficient solution by using the recession cone of the constraint set. For bicriteria problems, it is enough to employ a system of two regularity conditions. If the number of criteria exceeds two, a third regularity condition must be added to the system. The obtained results complement the above-mentioned remarkable theorem of Choo and are analyzed through several interesting examples, including those given by Hoa et al. (J Ind Manag Optim 1:477–486, 2005).

New results on proper efficiency for a class of vector optimization problems

ABSTRACT This paper presents two new theorems on Geoffrion's properly efficient solutions and seven examples illustrating their applications to linear fractional vector optimization problems with unbounded constraint sets. Provided that all the components of the objective function are properly fractional, the first theorem gives sufficient conditions for the efficient solution set to coincide with the Geoffrion properly efficient solution set. Admitting that the objective function can have some affine components, in the second theorem we give sufficient conditions for an efficient solution to be a Geoffrion's properly efficient solution. The recession cone of the constraint set, the derivatives of the scalar objective functions, but no tangent cone to the constraint set at the efficient point, are used in the second theorem.

A representation of generalized convex polyhedra and applications

ABSTRACTIt is well known that finite-dimensional polyhedral convex sets can be generated by finitely many points and finitely many directions. Representation formulas in this spirit are obtained for convex polyhedra and generalized convex polyhedra in locally convex Hausdorff topological vector spaces. Our results develop those of X. Y. Zheng (Set-Valued Anal. 2009;17:389–408), which were established in a Banach space setting. Applications of the representation formulas to proving solution existence theorems for generalized linear programming problems and generalized linear vector optimization problems are shown.

Affine Variational Inequalities on Normed Spaces

This paper studies infinite-dimensional affine variational inequalities on normed spaces. It is shown that infinite-dimensional quadratic programming problems and infinite-dimensional linear fractional vector optimization problems can be studied by using affine variational inequalities. We present two basic facts about infinite-dimensional affine variational inequalities: the Lagrange multiplier rule and the solution set decomposition.

Efficient solutions in generalized linear vector optimization

ABSTRACTThis paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized linear vector optimization problems is obtained. We also prove that the efficient solution set of a generalized linear vector optimization problem in a locally convex Hausdorff topological vector space is the union of finitely many generalized polyhedral convex sets and it is connected by line segments.

Connectedness of Solution Sets of Strong Vector Equilibrium Problems with an Application

In this paper, the connectedness of solution set of a strong vector equilibrium problem in a finite dimensional space, is investigated. Firstly, a nonconvex separation theorem is given, that is, a neither open nor closed set and a compact subset in a finite dimensional space can be strictly separated by a sublinear and strongly monotone function. Secondly, in terms of the nonconvex separation theorem, the union relation between the solution set of the strong vector equilibrium problem and the solution sets of a series of nonlinear scalar problems, is established. Under suitable assumptions, the connectedness and the path connectedness of the solution set of the strong vector equilibrium problem are obtained. In particular, we solve partly the question related to the path connectedness of the solution set of the strong vector equilibrium problem. The question is proposed by Han and Huang in the reference (J Optim Theory Appl, 2016. https://doi.org/10.1007/s10957-016-1032-9 ). Finally, as an application, we apply the main results to derive the connectedness of the solution set of a linear vector optimization problem.

Piecewise Linear Vector Optimization Problems on Locally Convex Hausdorff Topological Vector Spaces

Piecewise linear vector optimization problems in the locally convex Hausdorff topological vector space setting are considered in this paper. The efficient solution set of these problems are shown to be the unions of finitely many semi-closed generalized polyhedral convex sets. If, in addition, the problem is convex, then the efficient solution set and the weakly efficient solution set are the unions of finitely many generalized polyhedral convex sets and they are connected by line segments. Our results develop the preceding ones of Zheng and Yang (Sci. China Ser. A. 51, 1243–1256 2008), and Yang and Yen (J. Optim. Theory Appl. 147, 113–124 2010), which were established in the normed space setting.

A parametric simplex algorithm for linear vector optimization problems

In this paper, a parametric simplex algorithm for solving linear vector optimization problems (LVOPs) is presented. This algorithm can be seen as a variant of the multi-objective simplex (Evans-Steuer) algorithm [12]. Different from it, the proposed algorithm works in the parameter space and does not aim to find the set of all efficient solutions. Instead, it finds a solution in the sense of Loehne [16], that is, it finds a subset of efficient solutions that allows to generate the whole frontier. In that sense, it can also be seen as a generalization of the parametric self-dual simplex algorithm, which originally is designed for solving single objective linear optimization problems, and is modified to solve two objective bounded LVOPs with the positive orthant as the ordering cone in Ruszczynski and Vanderbei [21]. The algorithm proposed here works for any dimension, any solid pointed polyhedral ordering cone C and for bounded as well as unbounded problems. Numerical results are provided to compare the proposed algorithm with an objective space based LVOP algorithm (Benson algorithm in [13]), that also provides a solution in the sense of [16], and with Evans-Steuer algorithm [12]. The results show that for non-degenerate problems the proposed algorithm outperforms Benson algorithm and is on par with Evan-Steuer algorithm. For highly degenerate problems Benson's algorithm [13] excels the simplex-type algorithms; however, the parametric simplex algorithm is for these problems computationally much more efficient than Evans-Steuer algorithm.

The Adaptive Parameter Control Method and Linear Vector Optimization

The question of constructing a set of equidistant points, with a given small approximate distance, in the efficient and weakly efficient frontiers of a linear vector optimization problem of a general form, is considered in this paper. It is shown that the question can be solved by combining Pascoletti–Serafini’s scalarization method (1984) and Eichfelder’s adaptive parameter control method (2009) with a sensitivity analysis formula in linear programming, which was obtained by J. Gauvin (2001). Our investigation shows that one can avoid the strong second-order sufficient condition used by G. Eichfelder, which cannot be imposed on linear vector optimization problems.

On the solution of Large Scale Bi-Level Linear Vector Optimization Problems through TOPSIS

In this paper, we extend TOPSIS (Technique for Order Preference by Similarity Ideal Solution) for solving Large Scale Bi-level Linear Vector Optimization Problems (LS-BL-LVOP). In order to obtain a compromise ( satisfactory) solution to the LS-BL-LVOP problems using the proposed TOPSIS approach, a modified formulas for the distance function from the positive ideal solution (PIS ) and the distance function from the negative ideal solution (NIS) are proposed and modeled to include all the objective functions of both the first and the second levels. An interactive decision making algorithm for generating a compromise ( satisfactory) solution through TOPSIS approach is provided where the first level decision maker (FLDM) is asked to specify the relative importance of the objectives. Finally, a numerical example is given to clarify the main results developed in the paper.