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Criteria for efficiency in vector optimization

Author

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  • Francisco Vázquez
  • Hubertus Jongen
  • Vladimir Shikhman
  • Maxim Todorov

Abstract

We consider unconstrained finite dimensional multi-criteria optimization problems, where the objective functions are continuously differentiable. Motivated by previous work of Brosowski and da Silva (1994), we suggest a number of tests (TEST 1–4) to detect, whether a certain point is a locally (weakly) efficient solution for the underlying vector optimization problem or not. Our aim is to show: the points, at which none of the TESTs 1–4 can be applied, form a nowhere dense set in the state space. TESTs 1 and 2 are exactly those proposed by Brosowski and da Silva. TEST 3 deals with a local constant behavior of at least one of the objective functions. TEST 4 includes some conditions on the gradients of objective functions satisfied locally around the point of interest. It is formulated as a Conjecture. It is proven under additional assumptions on the objective functions, such as linear independence of the gradients, convexity or directional monotonicity. Copyright Springer-Verlag 2009

Suggested Citation

  • Francisco Vázquez & Hubertus Jongen & Vladimir Shikhman & Maxim Todorov, 2009. "Criteria for efficiency in vector optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(1), pages 35-46, August.
    • Handle: RePEc:spr:mathme:v:70:y:2009:i:1:p:35-46
    DOI: 10.1007/s00186-008-0230-0
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    Keywords

    Vector optimization; Efficiency criteria; Density; 49D39; 65F99; 15A39;
    All these keywords.

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