Criteria for efficiency in vector optimization

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.

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References

  1. Brosowski B, da Silva AR (1994) Simple tests for multi-criteria. OR Spectrum 16: 243–247

    MATH  Google Scholar 

  2. Demjanov VF, Malozemov VN (1974) Introduction to Minimax, translated from Russian by Louvish D. Wiley, New York

    Google Scholar 

  3. Floudas CA, Pardalos PM (2001) Encyclopedia of Optimization, vol III. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  4. Rockafellar RT, Wets RJ-B (1998) Variational Analysis. Springer, Berlin

    Google Scholar 

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Correspondence to Francisco Guerra Vázquez.

Additional information

This work was partially supported by grant 55681 of the CONACyT.

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Vázquez, F.G., Jongen, H.T., Shikhman, V. et al. Criteria for efficiency in vector optimization. Math Meth Oper Res 70, 35–46 (2009). https://doi.org/10.1007/s00186-008-0230-0

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Keywords

  • Vector optimization
  • Efficiency criteria
  • Density

Mathematics Subject Classification (2000)

  • 49D39
  • 65F99
  • 15A39