Advertisement

Mathematical Methods of Operations Research

, Volume 70, Issue 1, pp 35–46 | Cite as

Criteria for efficiency in vector optimization

  • Francisco Guerra Vázquez
  • Hubertus Th. Jongen
  • Vladimir Shikhman
  • Maxim Ivanov Todorov
Original Article

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.

Keywords

Vector optimization Efficiency criteria Density 

Mathematics Subject Classification (2000)

49D39 65F99 15A39 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Brosowski B, da Silva AR (1994) Simple tests for multi-criteria. OR Spectrum 16: 243–247zbMATHGoogle Scholar
  2. Demjanov VF, Malozemov VN (1974) Introduction to Minimax, translated from Russian by Louvish D. Wiley, New YorkGoogle Scholar
  3. Floudas CA, Pardalos PM (2001) Encyclopedia of Optimization, vol III. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
  4. Rockafellar RT, Wets RJ-B (1998) Variational Analysis. Springer, BerlinzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Francisco Guerra Vázquez
    • 1
  • Hubertus Th. Jongen
    • 2
  • Vladimir Shikhman
    • 2
  • Maxim Ivanov Todorov
    • 1
  1. 1.Department of Physics and MathematicsUDLASan Andrés CholulaMexico
  2. 2.Department of Mathematics – CRWTH Aachen UniversityAachenGermany

Personalised recommendations