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Variational Inequalities in Vector Optimization

  • G. P. Crespi
  • I. Ginchev
  • M. Rocca
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 79)

Abstract

In this paper we investigate the links among generalized scalar variational inequalities of differential type, vector variational inequalities and vector optimization problems. The considered scalar variational inequalities are obtained through a nonlinear scalarization by means of the so called “oriented distance” function [14,15].

In the case of Stampacchia-type variational inequalities, the solutions of the proposed ones coincide with the solutions of the vector variational inequalities introduced by Giannessi [8]. For Minty-type variational inequalities, analogous coincidence happens under convexity hypotheses. Furthermore, the considered variational inequalities reveal useful in filling a gap between scalar and vector variational inequalities. Namely, in the scalar case Minty variational inequalities of differential type represent a sufficient optimality condition without additional assumptions, while in the vector case the convexity hypothesis is needed. Moreover it is shown that vector functions admitting a solution of the proposed Minty variational inequality enjoy some well-posedness properties, analogously to the scalar case [4].

Keywords

Variational Inequality Strong Solution Vector Optimization Vector Optimization Problem Vector Variational Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • G. P. Crespi
    • 1
  • I. Ginchev
    • 2
  • M. Rocca
    • 3
  1. 1.Faculty of EconomicsUniversity of Valle d’AostaAostaItaly
  2. 2.Dept. of MathematicsTechnical University of VarnaVarnaBulgaria
  3. 3.Dept. of EconomicsUniversity of InsubriaVareseItaly

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