123 research outputs found

    The proximal point method for locally lipschitz functions in multiobjective optimization with application to the compromise problem

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    This paper studies the constrained multiobjective optimization problem of finding Pareto critical points of vector-valued functions. The proximal point method considered by Bonnel, Iusem, and Svaiter [SIAM J. Optim., 15 (2005), pp. 953–970] is extended to locally Lipschitz functions in the finite dimensional multiobjective setting. To this end, a new (scalarization-free) approach for convergence analysis of the method is proposed where the first-order optimality condition of the scalarized problem is replaced by a necessary condition for weak Pareto points of a multiobjective problem. As a consequence, this has allowed us to consider the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. This is very important for applications; for example, to extend to a dynamic setting the famous compromise problem in management sciences and game theory.Fundação de Amparo à Pesquisa do Estado de GoiásConselho Nacional de Desenvolvimento Científico e TecnológicoCoordenação de Aperfeiçoamento de Pessoal de Nivel SuperiorMinisterio de Economía y CompetitividadAgence nationale de la recherch

    Well-posedness in vector optimization and scalarization results

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    In this paper, we give a survey on well-posedness notions of Tykhonov's type for vector optimization problems and the links between them with respect to the classification proposed by Miglierina, Molho and Rocca in [33]. We consider also the notions of extended well-posedness introduced by X.X. Huang ([19],[20]) in the nonparametric case to complete the hierchical structure characterizing these concepts. Finally we propose a review of some theoretical results in vector optimization mainly related to different notions of scalarizing functions, linear and nonlinear, introduced in the last decades, to simplify the study of various well-posedness properties.

    Well-posedness and scalarization in vector optimization

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    In this paper we study several existing notions of well-posedness for vector optimization problems. We distinguish them into two classes and we establish the hierarchical structure of their relationships. Moreover, we relate vector well-posedness and well-posedness of an appropriate scalarization. This approach allows us to show that, under some compactness assumption, quasiconvex problems are well-posed.well-posedness, vector optimization problems, nonlinear scalarization, generalized convexity.

    Isolated minimizers, proper efficiency and stability for C0,1 constrained vector optimization problems

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    In this paper we consider the vector optimization problem minC f(x), g(x) 2 -K, where f : Rn ! Rm and g : Rn Rp are C0,1 functions and C Rm and K Rp are closed convex cones. We give several notions of solutions (efficiency concepts), among them the notion of a properly efficient point (p-minimizer) of order k and the notion of an isolated minimizer of order k. We show that each isolated minimizer of order k > = 1 is a p-minimizer of order k. The possible reversal of this statement in the case k = 1 is the main subject of the investigation. For this study we apply some first order necessary and sufficient conditions in terms of Dini derivatives. We show that the given optimality conditions are important to solve the posed problem, and a satisfactory solution leads to two approaches toward efficiency concepts, called respectively sense I and sense II concepts. Relations between sense I and sense II isolated minimizers and p-minimizers are obtained. In particular, we are concerned in the stability properties of the p-minimizers and the isolated minimizers. By stability, we mean that they still remain the same type of solutions under small perturbations of the problem data. We show that the p-minimizers are stable under perturbations of the cones, while the isolated minimizers are stable under perturbations both of the cones and the functions in the data set. Further, we show that the sense I concepts are stable under perturbations of the objective data, while the sense II concepts are stable under perturbations both of the objective and the constraints.Vector optimization, Locally Lipschitz data, Properly efficient points, Isolated minimizers, Optimality conditions, Stability.

    A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions

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    In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving non-smooth convex objective functions. Our approach is in the line of a previous work where was considered the case of convex di erentiable objective functions. Based on the Yosida regularization of the subdi erential operators involved in the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and the convergence of trajectories to weak Pareto minima. This approach provides a dynamical endogenous weighting of the objective functions. Applications are given to cooperative games, inverse problems, and numerical multiobjective optimization

    First-Order Conditions for C0,1 Constrained vector optimization

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    For a Fritz John type vector optimization problem with C0,1 data we define different type of solutions, give their scalar characterizations applying the so called oriented distance, and give necessary and sufficient first order optimality conditions in terms of the Dini derivative. While establishing the sufficiency, we introduce new type of efficient points referred to as isolated minimizers of first order, and show their relation to properly efficient points. More precisely, the obtained necessary conditions are necessary for weakly efficiency, and the sufficient conditions are both sufficient and necessary for a point to be an isolated minimizer of first order.vector optimization, nonsmooth optimization, C0,1 functions, Dini derivatives, first-order optimality conditions, lagrange multipliers

    Optimality conditions for approximate solutions of set-valued optimization problems in real linear spaces

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    In this paper, we deal with optimization problems without assuming any topology. We study approximate efficiency and Q- Henig proper efficiency for the setvalued vector optimization problems, where Q is not necessarily convex. We use scalarization approaches based on nonconvex separation function to present some necessary and sufficient conditions for approximate (proper and weak) efficient solutions.Publisher's Versio

    Scalarization and sensitivity analysis in Vector Optimization. The linear case.

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    In this paper we consider a vector optimization problem; we present some scalarization techniques for finding all the vector optimal points of this problem and we discuss the relationships between these methods. Moreover, in the linear case, the study of dual variables is carried on by means of sensitivity analysis and also by a parametric approach. We also give an interpretation of the dual variables as marginal rates of substitution of an objective function with respect to another one, and of an objective function with respect to a constraint.Vector Optimization, Image Space, Separation, Scalarization, Shadow Prices
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