In this paper, we propose a modification of Benson's algorithm for solving multiobjective linear programmes in objective space in order to approximate the true nondominated set. We first summarize Benson's ori...
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In this paper, we propose a modification of Benson's algorithm for solving multiobjective linear programmes in objective space in order to approximate the true nondominated set. We first summarize Benson's original algorithm and propose some small changes to improve computational performance. We then introduce our approximation version of the algorithm, which computes an inner and an outer approximation of the nondominated set. We prove that the inner approximation provides a set of epsilon-nondominated points. This work is motivated by an application, the beam intensity optimization problem of radiotherapy treatment planning. This problem can be formulated as a multiobjective linear programme with three objectives. The constraint matrix of the problem relies on the calculation of dose deposited in tissue. Since this calculation is always imprecise solving the MOLP exactly is not necessary in practice. With our algorithm we solve the problem approximately within a specified accuracy in objective space. We present results on four clinical cancer cases that clearly illustrate the advantages of our method.
We consider the usage of evolutionary algorithms for multiobjective programming (MOP), i.e. for decision problems with alternatives taken from a real-valued vector space and evaluated according to a vector-valued obje...
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We consider the usage of evolutionary algorithms for multiobjective programming (MOP), i.e. for decision problems with alternatives taken from a real-valued vector space and evaluated according to a vector-valued objective function. Selection mechanisms, possibilities of temporary fitness deterioration, and problems of unreachable alternatives for such multiobjective evolutionary algorithms (MOEAs) are studied. Theoretical properties of MOEAs such as stochastic convergence with probability 1 are analyzed. (C) 1999 Elsevier Science B.V. All rights reserved.
In this paper, I introduce two kinds of epsilon-subgradients epsilon-Pareto subgradients and epsilon-Benson proper subgradientsfor set-valued maps. I give sufficient conditions for the existence of such epsilon-subgra...
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In this paper, I introduce two kinds of epsilon-subgradients epsilon-Pareto subgradients and epsilon-Benson proper subgradientsfor set-valued maps. I give sufficient conditions for the existence of such epsilon-subgradients. A generalized epsilon-Moreau-Rockafellar type theorem for epsilon-Benson proper subdifferentials of set-valued maps is formulated and applied to establish epsilon-optimality conditions for epsilon-Benson proper efficientsolutions of vector optimization problems with set-valued maps. Scalarization theorems for such a problem are also obtained.
Based on a general framework for a class of (rho, eta, A)-invex n-set functions, some results on the epsilon-optimality conditions for multiple objective fractional programming problems are established. The obtained r...
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Based on a general framework for a class of (rho, eta, A)-invex n-set functions, some results on the epsilon-optimality conditions for multiple objective fractional programming problems are established. The obtained results are not only general in nature, but also unify results on generalized invexity to the context of multiple fractional programming. (C) 2012 Elsevier Inc. All rights reserved.
In this note, we show that for linear fractional vector optimization problems with bounded constraint sets there is no difference between the epsilon-efficiency and the epsilon proper efficiency.
In this note, we show that for linear fractional vector optimization problems with bounded constraint sets there is no difference between the epsilon-efficiency and the epsilon proper efficiency.
In this paper, a scalarization result of epsilon-weak efficientsolution for a vector equilibrium problem (VEP) is given. Using this scalarization result, the connectedness of epsilon-weak efficient and epsilon-effici...
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In this paper, a scalarization result of epsilon-weak efficientsolution for a vector equilibrium problem (VEP) is given. Using this scalarization result, the connectedness of epsilon-weak efficient and epsilon-efficient solutions sets for the VEPs are proved under some suitable conditions in real Hausdorff topological vector spaces. The main results presented in this paper improve and generalize some known results in the literature.
In this paper, a well-known concept of epsilon-efficient solution due to Kutateladze is studied, in order to approximate the weak efficientsolutions of vector optimization problems. In particular, it is proved that t...
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ISBN:
(纸本)9783319181615;9783319181608
In this paper, a well-known concept of epsilon-efficient solution due to Kutateladze is studied, in order to approximate the weak efficientsolutions of vector optimization problems. In particular, it is proved that the limit, in the Painleve-Kuratowski sense, of the epsilon-efficient sets when the precision epsilon tends to zero is the set of weak efficientsolutions of the problem. Moreover, several nonlinear scalarization results are derived to characterize the epsilon-efficient solutions in terms of approximate solutions of scalar optimization problems. Finally, the obtained results are applied not only to propose a kind of penalization scheme for Kutateladze's approximate solutions of a cone constrained convex vector optimization problem but also to characterize epsilon-efficient solutions of convex multiobjective problems with inequality constraints via multiplier rules.
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