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On Approximate Solutions in Vector Optimization Problems Via Scalarization

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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.

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Authors and Affiliations

  1. Departamento de Matem´tica Aplicada, E.T.S.I. Inform´tica, Universidad de Valla-dolid, Edificio de Tecnologías de la Información y las Telecomunicaciones, Campus Miguel Delibes, s/n, 47011, Valladolid, Spain

    César Gutiérrez

  2. Departamento de Matemática Aplicada, E.T.S.I. Industriales, UNED, Ciudad Universitaria, c/ Juan del Rosal, 12, 28040, Madrid, Spain

    Bienvenido Jiménez & Vicente Novo

Authors
  1. César Gutiérrez
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  2. Bienvenido Jiménez
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  3. Vicente Novo
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Corresponding author

Correspondence to César Gutiérrez.

Additional information

AMS Classification:90C29, 49M37

This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194.

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Gutiérrez, C., Jiménez, B. & Novo, V. On Approximate Solutions in Vector Optimization Problems Via Scalarization. Comput Optim Applic 35, 305–324 (2006). https://doi.org/10.1007/s10589-006-8718-0

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  • DOI: https://doi.org/10.1007/s10589-006-8718-0

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