Some Existence Results and Stability in Multi Objective Optimization

  • Roberto Lucchetti
Part of the International Centre for Mechanical Sciences book series (CISM, volume 289)


If someone is curious about multiobjective optimization, and enters in a library to look for works in the argument, he can find an enormous amount of things, results, applications, references. For this and other reasons, it is quite obvious that the results given here are a very little part of the known theorems, mainly in the existence. So, no claims to be complete. We shall deal with a topological vector space X and a cone C ⊂ ℝm that is always assumed closed convex, pointed (C ∩ − C = {0}) and with nonempty interior. Moreover it is given a function f: X → ℝm to be maximized with respect to the ordering induced by the cone, on a general constraint set A. We are interested in existence theorems, namely the non-emptiness of the set . At first, observe that, if C1 ⊂ C2, then : this means that, in particular, the non emptiness of sC guarantees existence of solutions for the weak optimization, namely of the points . In the sequel we write s for sC and W for the weak solutions. In the scalar case, i.e. when m = 1 the most celebrated theorem says that, if f is upper semicontinuous (u.s.c.) and there is in A a relatively compact maximizing sequence, then there is a solution for the problem. We want to present here an analogous simple result, so we have to clarify the meaning of “maximizing sequences” and “upper semicontinuity”. We shall make use of the following notations:


Multi Objective Optimization Existence Result Multiobjective Optimization Topological Vector Space Scalar Case 
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Copyright information

© Springer-Verlag Wien 1985

Authors and Affiliations

  • Roberto Lucchetti
    • 1
  1. 1.Istituto di MatematicaUniversity of GenoaItaly

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