Let a polyhedral convex set be given by a finite number of linear inequalities and consider the problem to project this set onto a subspace. This problem, called polyhedral projection problem, is shown to be equivalen...
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Let a polyhedral convex set be given by a finite number of linear inequalities and consider the problem to project this set onto a subspace. This problem, called polyhedral projection problem, is shown to be equivalent to multiple objective linearprogramming. The number of objectives of the multiple objective linear program is by one higher than the dimension of the projected polyhedron. The result implies that an arbitrary vectorlinear program (with arbitrary polyhedral ordering cone) can be solved by solving a multiple objective linear program (i.e. a vectorlinear program with the standard ordering cone) with one additional objective space dimension.
We provide a solution method for the polyhedral convex set optimization problem. that is, the problem to minimize a set-valued maping F: (RRq)-R-n paired right arrows with polyhedral convex graph with respect to a set...
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We provide a solution method for the polyhedral convex set optimization problem. that is, the problem to minimize a set-valued maping F: (RRq)-R-n paired right arrows with polyhedral convex graph with respect to a set ordering relation which is gelerated by a polyhedral convex cone C subset of R-q. The methods proven to be correct and finite without any further assumption to the problem.
For a given polyhedral convex set-valued mapping we define a polyhedral convex cone which we call the natural ordering cone. We show that the solution behaviour of a polyhedral convex set optimization problem can be c...
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For a given polyhedral convex set-valued mapping we define a polyhedral convex cone which we call the natural ordering cone. We show that the solution behaviour of a polyhedral convex set optimization problem can be characterized by this cone. Under appropriate assumptions, the natural ordering cone is proven to be the smallest ordering cone which makes a polyhedral convex set optimization problem solvable.
Bensolve is an open source implementation of Benson's algorithm and its dual variant. Both algorithms compute primal and dual solutions of vectorlinear programs (VLP), which include the subclass of multiple objec...
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Bensolve is an open source implementation of Benson's algorithm and its dual variant. Both algorithms compute primal and dual solutions of vectorlinear programs (VLP), which include the subclass of multiple objective linear programs (MOLP). The recent version of Bensolve can treat arbitrary vectorlinear programs whose upper image does not contain lines. This article surveys the theoretical background of the implementation. In particular, the role of VLP duality for the implementation is pointed out. Some numerical examples are provided. In contrast to the existing literature we consider a less restrictive class of vectorlinear programs. (C) 2016 Elsevier B.V. All rights reserved.
In this paper, we deal with a multicriteria competitive Markov decision process. In the decision process there are two decision makers with a competitive behaviour, so they are usually called players. Their rewards ar...
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In this paper, we deal with a multicriteria competitive Markov decision process. In the decision process there are two decision makers with a competitive behaviour, so they are usually called players. Their rewards are coupled because depend on the actions chosen by both players in each state of the process. We propose as solution of this game the set of Pareto-optimal security strategies for any of the players in the original problem. We show that this solution set can be obtained as the efficient solution set of a multicriteria linearprogramming problem.
In this paper, a multiple-objective linear problem is derived from a zero-sum multicriteria matrix game. It is shown that the set of efficient solutions of this problem coincides with the set of Pareto-optimal securit...
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In this paper, a multiple-objective linear problem is derived from a zero-sum multicriteria matrix game. It is shown that the set of efficient solutions of this problem coincides with the set of Pareto-optimal security strategies (POSS) for one of the players in the original game. This approach emphasizes the existing similarities between the scalar and multicriteria matrix games, because in both cases linearprogramming can be used to solve the problems. It also leads to different scalarizations which are alternative ways to obtain the set of all POSS. The concept of ideal strategy for a player is introduced, and it is established that a pair of Pareto saddle-point strategies exists if both players have ideal strategies. Several examples are included to illustrate the results in the paper.
In this paper we present the Ordered Multiobjective linearprogramming, where the utility function in x, is given by the k-th lowest value of the vector (c 1x, c2x,...,cpx). We study the problem in the case of partial...
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