{
  "id": "1602.03367",
  "version": "v1",
  "published": "2016-02-10T13:33:57.000Z",
  "updated": "2016-02-10T13:33:57.000Z",
  "title": "Characterizing weak solutions for vector optimization problems",
  "authors": [
    "Nguyen Dinh",
    "Miguel A. Goberna",
    "Dang H. Long",
    "Marco A. López"
  ],
  "comment": "23 pages, 0 figures",
  "categories": [
    "math.OC"
  ],
  "abstract": "This paper provides characterizations of the weak solutions of optimization problems where a given vector function $F,$ from a decision space $X$ to an objective space $Y$, is \"minimized\" on the set of elements $x\\in C$ (where $C\\subset X$ is a given nonempty constraint set), satisfying $G\\left( x\\right) \\leqq_{S}0_{Z},$ where $G$ is another given vector function from $X $ to a constraint space $Z$ with positive cone $S$. The three spaces $X,Y,$ and $Z$ are locally convex Hausdorff topological vector spaces, with $Y$ and $Z$ partially ordered by two convex cones $K$ and $S,$ respectively, and enlarged with a greatest and a smallest element. In order to get suitable versions of the Farkas lemma allowing to obtain optimality conditions expressed in terms of the data, the triplet $\\left( F,G,C\\right) ,$ we use non-asymptotic representations of the $K-$epigraph of the conjugate function of $F+I_{A},$ where $I_{A}$ denotes the indicator function of the feasible set $A,$ that is, the function associating the zero vector of $Y$ to any element of $A$ and the greatest element of $Y$ to any element of $X\\diagdown A.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-10T13:33:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58E17",
      "90C29",
      "90C46",
      "49N15"
    ],
    "keywords": [
      "vector optimization problems",
      "characterizing weak solutions",
      "vector function",
      "convex hausdorff topological vector spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160203367D"
    }
  }
}