Gabriele Eichfelder

On classes of set optimization problems which are reducible to vector optimization problems and its impact on numerical test instances

  • Gabriele Eichfelder
  • Tobias Gerlach
    • Categories Multi-Criteria Optimization, Nonlinear Optimization Tags , , , ,

    Set optimization with the set approach has recently gained increasing interest due to its practical relevance. In this problem class one studies optimization problems with a set-valued objective map and defines optimality based on a direct comparison of the images of the objective function, which are sets here. Meanwhile, in the literature a wide range … Read more

    Set approach for set optimization with variable ordering structures

  • Gabriele Eichfelder
  • Maria Pilecka
    • Categories Infinite Dimensional Optimization, Multi-Criteria Optimization, Other Topics

    This paper aims at combining variable ordering structures with set relations in set optimization, which have been defined using the constant ordering cone before. Since the purpose is to connect these two important approaches in set optimization, we do not restrict our considerations to one certain relation. Conversely, we provide the reader with many new … Read more

    On the effects of combining objectives in multi-objective optimization

  • Stephan Dempe
  • Gabriele Eichfelder
  • Jörg Fliege
    • Categories Multi-Criteria Optimization

    In multi-objective optimization, one considers optimization problems with more than one objective function, and in general these objectives conflict each other. As the solution set of a multiobjective problem is often rather large and contains points of no interest to the decision-maker, strategies are sought that reduce the size of the solution set. One such … Read more

    Characterization of properly optimal elements with variable ordering structures

  • Gabriele Eichfelder
  • Tobias Gerlach
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization Tags , , ,

    In vector optimization with a variable ordering structure the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. In recent publications it was started to develop a comprehensive theory for these vector optimization problems. Thereby also notions of proper efficiency … Read more

    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

    Erratum to: On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets” [Optim. Letters, 2012]

  • Peter J.C. Dickinson
  • Gabriele Eichfelder
  • Janez Povh
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    In this paper, an erratum is provided to the article “\emph{On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets}”, published in Optim.\ Letters, 2012. Due to precise observation of the first author, it has been found that the proof of Lemma 9 has a nontrivial gap, and consequently the main result (Theorem … Read more

    On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets

  • Gabriele Eichfelder
  • Janez Povh
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming, Quadratic Programming Tags , , ,

    In the paper we prove that any nonconvex quadratic problem over some set $K\subset \mathbb{R}^n$ with additional linear and binary constraints can be rewritten as linear problem over the cone, dual to the cone of K-semidefinite matrices. We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and … Read more

    On reformulations of nonconvex quadratic programs over convex cones by set-semidefinite constraints

  • Gabriele Eichfelder
  • Janez Povh
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming

    The well-known result stating that any non-convex quadratic problem over the nonnegative orthant with some additional linear and binary constraints can be rewritten as linear problem over the cone of completely positive matrices (Burer, 2009) is generalized by replacing the nonnegative orthant with an arbitrary closed convex cone. This set-semidefinite representation result implies new semidefinite … Read more

    Copositivity detection by difference-of-convex decomposition and omega-subdivision

  • Immanuel M. Bomze
  • Gabriele Eichfelder
    • Categories Linear, Cone and Semidefinite Programming Tags

    We present three new copositivity tests based upon difference-of-convex (d.c.) decompositions, and combine them to a branch-and-bound algorithm of $\omega$-subdivision type. The tests employ LP or convex QP techniques, but also can be used heuristically using appropriate test points. We also discuss the selection of efficient d.c.~decompositions and propose some preprocessing ideas based on the … Read more