vector optimization

Properly optimal elements in vector optimization with variable ordering structures

  • Gabriele Eichfelder
  • Refail Kasimbeyli
    • Categories Multi-Criteria Optimization, Nonlinear Optimization Tags , , ,

    In this paper, proper optimality concepts in vector optimization with variable ordering structures are introduced for the first time and characterization results via scalarizations are given. New type of scalarizing functionals are presented and their properties are discussed. The scalarization approach suggested in the paper does not require convexity and boundedness conditions. CitationPreprint of the … Read more

    A Generalization of a Theorem of Arrow, Barankin and Blackwell to a Nonconvex Case

  • Refail Kasimbeyli
  • Musa Mammadov
    • Categories Global Optimization, Global Optimization Theory, Multi-Criteria Optimization Tags , , , , , ,

    The paper presents a generalization of a known density theorem of Arrow, Barankin, and Blackwell for properly efficient points defined as support points of sets with respect to monotonically increasing sublinear functions. This result is shown to hold for nonconvex sets of a reflexive Banach space partially ordered by a Bishop–Phelps cone. CitationDepartment of Industrial … Read more

    POST-PARETO ANALYSIS FOR MULTIOBJECTIVE PARABOLIC CONTROL SYSTEMS

  • Henri Bonnel
    • Categories Distributed Control, Multi-Criteria Optimization Tags , , , ,

    In this paper is presented the problem of optimizing a functional over a Pareto control set associated with a convex multiobjective control problem in Hilbert spaces, namely parabolic system. This approach generalizes for this setting some results obtained in finite dimensions. Some examples are presented. General optimality results are obtained, and a special attention is … Read more

    A quadratically convergent Newton method for vector optimization

  • L. M. Graña Drummond
  • Benar Fux Svaiter
  • Fernanda M. P. Raupp
    • Categories Convex Optimization, Multi-Criteria Optimization, Nonlinear Optimization Tags , , ,

    We propose a Newton method for solving smooth unconstrained vector optimization problems under partial orders induced by general closed convex pointed cones. The method extends the one proposed by Fliege, Grana Drummond and Svaiter for multicriteria, which in turn is an extension of the classical Newton method for scalar optimization. The steplength is chosen by … Read more

    Inexact projected gradient method for vector optimization

  • Ellen H. Fukuda
  • L. M. Graña Drummond
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization Tags , , ,

    In this work, we propose an inexact projected gradient-like method for solving smooth constrained vector optimization problems. In the unconstrained case, we retrieve the steepest descent method introduced by Graña Drummond and Svaiter. In the constrained setting, the method we present extends the exact one proposed by Graña Drummond and Iusem, since it admits relative … Read more

    An Interior Proximal Method in Vector Optimization

  • Paulo Roberto Oliveira
  • Kely Diana Villacorta
    • Categories Multi-Criteria Optimization Tags , ,

    This paper studies the vector optimization problem of finding weakly ef- ficient points for maps from Rn to Rm, with respect to the partial order induced by a closed, convex, and pointed cone C ⊂ Rm, with nonempty inte- rior. We develop for this problem an extension of the proximal point method for scalar-valued convex … Read more

    On the convergence of the projected gradient method for vector optimization

  • Ellen H. Fukuda
  • L. M. Graña Drummond
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization Tags , , ,

    In 2004, Graña Drummond and Iusem proposed an extension of the projected gradient method for constrained vector optimization problems. In that method, an Armijo-like rule, implemented with a backtracking procedure, was used in order to determine the steplengths. The authors just showed stationarity of all cluster points and, for another version of the algorithm (with … Read more

    Dido’s Problem and Pareto Optimality

  • S. S. Kutateladze
    • Categories Convex Optimization, Multi-Criteria Optimization Tags , ,

    Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body~$\mathfrak x$, we try to maximize the volume of~$\mathfrak x$ and minimize the width of~$\mathfrak x$ simultaneously. These problems are addressed … Read more

    THE EKELAND VARIATIONAL PRINCIPLE FOR HENIG PROPER MINIMIZERS AND SUPER MINIMIZERS

  • Xuan Duc Ha Truong
    • Categories Multi-Criteria Optimization, Nonsmooth Optimization Tags , , , , , , ,

    In this paper we consider, for the first time, approximate Henig proper minimizers and approximate super minimizers of a set-valued map F with values in a partially ordered vector space and formulate two versions of the Ekeland variational principle for these points involving coderivatives in the senses of Ioffe, Clarke and Mordukhovich. As applications we … Read more

    An Efficient Interior-Point Method for Convex Multicriteria Optimization Problems

  • Jörg Fliege
    • Categories Convex Optimization, Multi-Criteria Optimization Tags , , , ,

    In multicriteria optimization, several objective functions, conflicting with each other, have to be minimized simultaneously. We propose a new efficient method for approximating the solution set of a multiobjective programming problem, where the objective functions involved are arbitary convex functions and the set of feasible points is convex. The method is based on generating warm-start … Read more