A multiobjective programming algorithm may find multiple nondominated solutions. If these solutions are scattered more uniformly over the Pareto frontier in the objective space, they are more different choices and so ...
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A multiobjective programming algorithm may find multiple nondominated solutions. If these solutions are scattered more uniformly over the Pareto frontier in the objective space, they are more different choices and so their quality is better. In this paper, we propose a quality measure called U-measure to measure the uniformity of a given set of nondominated solutions over the Pareto frontier. This frontier is a nonlinear hyper-surface. We measure the uniformity over this hyper-surface in three main steps: 1) determine the domains of the Pareto frontier over which uniformity is measured, 2) determine the nearest neighbors of each solution in the objective space, and 3) compute the discrepancy among the distances between nearest neighbors. The U-measure is equal to this discrepancy where a smaller discrepancy indicates a better uniformity. We can apply the U-measure to complement the other quality measures so that we can evaluate and compare multiobjective programming algorithms from different perspectives.
This paper proposes a new algorithm to solve nonsmooth multiobjective programming. The algorithm is a descent direction method to obtain the critical point (a necessary condition for Pareto optimality). We analyze bot...
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This paper proposes a new algorithm to solve nonsmooth multiobjective programming. The algorithm is a descent direction method to obtain the critical point (a necessary condition for Pareto optimality). We analyze both global and local convergence results under some assumptions. Numerical tests are also given. (C) 2014 Elsevier B.V. All rights reserved.
In this paper a vector optimization problem (VOP) is considered where each component of objective and constraint function involves a term containing support function of a compact convex set. Weak and strong Kuhn-Tucke...
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In this paper a vector optimization problem (VOP) is considered where each component of objective and constraint function involves a term containing support function of a compact convex set. Weak and strong Kuhn-Tucker necessary optimality conditions for the problem are obtained under suitable constraint qualifications. Necessary and sufficient conditions are proved for a critical point to be a weak efficient or an efficient solution of the problem (VOP) assuming that the functions belong to different classes of pseudoinvex functions. Two Mond Weir type dual problems are considered for (VOP) and duality results are established.
In this paper, first and second order sufficient conditions are established for strict local Pareto minima of orders 1 and 2 to multiobjective optimization problems with an arbitrary feasible set and a twice different...
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In this paper, first and second order sufficient conditions are established for strict local Pareto minima of orders 1 and 2 to multiobjective optimization problems with an arbitrary feasible set and a twice differentiable objective function are provided. For this aim, the concept of support function to a multiobjective problem is introduced, so that the scalar case in particular is contained. The obtained results generalize the classical ones of this case. Furthermore, particularizing to a feasible set defined by equality and inequality constraints, first and second order optimality conditions in primal form as well as dual form (by means of a Lagrange multiplier rule) are obtained.
To facilitate the evaluation of tradeoffs and the articulation of preferences in multiple criteria decision-making, a multiobjective decomposition scheme is proposed that restructures the original problem as a collect...
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To facilitate the evaluation of tradeoffs and the articulation of preferences in multiple criteria decision-making, a multiobjective decomposition scheme is proposed that restructures the original problem as a collection of smaller-sized subproblems with only subsets of the original criteria. A priori preferences on objective tradeoffs are integrated into this process by modifying the ordinary Pareto order by more general domination cones, and decision makers are supported by an interactive decision-making procedure to coordinate any remaining tradeoffs using concepts of approximate efficiency. A theoretical foundation for this method is provided, and an illustrative application to multiobjective portfolio optimization is described in detail. (C) 2009 Elsevier B.V. All rights reserved.
In this paper, we revisit one of the most important scalarization techniques used in multiobjective programming, the epsilon-constraint method. We summarize the method and point out some weaknesses, namely the lack of...
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In this paper, we revisit one of the most important scalarization techniques used in multiobjective programming, the epsilon-constraint method. We summarize the method and point out some weaknesses, namely the lack of easy-to-check conditions for properly efficient solutions and the inflexibility of the constraints. We present two modifications that address these weaknesses by first including slack variables in the formulation and second elasticizing the constraints and including surplus variables. We prove results on (weakly, properly) efficient solutions. The improved epsilon-constraint method that we propose combines both modifications.
In the present paper, we consider a nondifferentiable multiobjective programming problem with support functions and locally Lipschitz functions. Several sufficient optimality conditions are discussed for a strict mini...
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In the present paper, we consider a nondifferentiable multiobjective programming problem with support functions and locally Lipschitz functions. Several sufficient optimality conditions are discussed for a strict minimizer of a nondifferentiable multiobjective programming problem under strong invexity and its generalizations of order sigma. Weak and strong duality theorems are established for a Mond-Weir type dual.
We examine new second-order necessary conditions and sufficient conditions which characterize nondominated solutions of a generalized constrained multiobjective programming problem. The vector-valued criterion functio...
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We examine new second-order necessary conditions and sufficient conditions which characterize nondominated solutions of a generalized constrained multiobjective programming problem. The vector-valued criterion function as well as constraint functions are supposed to be from the class C1,1. Second-order optimality conditions for local Pareto solutions are derived as a special case.
In equitable multiobjective optimization all the objectives are uniformly optimized, but in some cases the decision maker believes that some of them should be uniformly optimized. To solve this problem in this paper, ...
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In equitable multiobjective optimization all the objectives are uniformly optimized, but in some cases the decision maker believes that some of them should be uniformly optimized. To solve this problem in this paper, the original problem is decomposed into a collection of smaller subproblems, according to the decision maker, and then the subproblems are solved by the concept of equitable efficiency. Furthermore, by using the concept of P-equitable efficiency two models are presented to coordinate equitably efficient solutions of subproblems. (C) 2014 Elsevier B.V. All rights reserved.
In this paper, optimality for multiobjective programming problems having invex objective and constraint functions (with respect to the same function eta) is considered. An equivalent vector programming problem is cons...
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In this paper, optimality for multiobjective programming problems having invex objective and constraint functions (with respect to the same function eta) is considered. An equivalent vector programming problem is constructed by a modification of the objective function. Furthermore, an eta-Lagrange function is introduced for a constructed multiobjective problem and modified saddle point results are presented.
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