On Approximate Solutions in Vector Optimization Problems Via Scalarization

  • Authors:

  • César Gutiérrez;Bienvenido Jiménez;Vicente Novo

  • Affiliations:

  • Departamento de Matemática Aplicada, E.T.S.I. Informática, Universidad de Valla-dolid, Edificio de Tecnologías de la Información y las Telecomunicaciones, Valladolid, Spain 470 ...;Departamento de Matemática Aplicada, E.T.S.I. Industriales, UNED, Ciudad Universitaria, Madrid, Spain 28040;Departamento de Matemática Aplicada, E.T.S.I. Industriales, UNED, Ciudad Universitaria, Madrid, Spain 28040

  • Venue:
  • Computational Optimization and Applications
  • Year:
  • 2006

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Abstract

This work deals with approximate solutions in vector optimization problems. These solutions frequently appear when an iterative algorithm is used to solve a vector optimization problem. We consider a concept of approximate efficiency introduced by Kutateladze and widely used in the literature to study this kind of solutions. Necessary and sufficient conditions for Kutateladze's approximate solutions are given through scalarization, in such a way that these points are approximate solutions for a scalar optimization problem. Necessary conditions are obtained by using gauge functionals while monotone functionals are considered to attain sufficient conditions. Two properties are then introduced to describe the idea of parametric representation of the approximate efficient set. Finally, through scalarization, characterizations and parametric representations for the set of approximate solutions in convex and nonconvex vector optimization problems are proved and the obtained results are applied to Pareto problems.