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Points of Efficiency in Vector Optimization with Increasing-along-rays Property and Minty Variational Inequalities

In: Generalized Convexity and Related Topics

Author

Listed:
  • Giovanni P. Crespi

    (Université de la Vallé d’Aoste)

  • Ivan Ginchev

    (Technical University of Varna)

  • Matteo Rocca

    (University of Insubria)

Abstract

Summary Minty variational inequalities are studied as a tool for vector optimization. Instead of focusing on vector inequalities, we propose an approach through scalarization which allows to construct a proper variational inequality type problem to study any concept of efficiency in vector optimization. This general scheme gives an easy and consistent extension of scalar results, providing also a notion of increasing along rays vector function. This class of generalized convex functions seems to be intimately related to the existence of solutions to a Minty variational inequality in the scalar case, we now extend this fact to vector case. Finally, to prove a reversal of the main theorem, generalized quasiconvexity is considered and the notion of *-quasiconvexity plays a crucial role to extend scalar evidences. This class of functions, indeed, guarantees a Minty-type variational inequality is a necessary and sufficient optimality condition for several kind of efficient solution.

Suggested Citation

  • Giovanni P. Crespi & Ivan Ginchev & Matteo Rocca, 2007. "Points of Efficiency in Vector Optimization with Increasing-along-rays Property and Minty Variational Inequalities," Lecture Notes in Economics and Mathematical Systems, in: Generalized Convexity and Related Topics, pages 209-226, Springer.
    • Handle: RePEc:spr:lnechp:978-3-540-37007-9_12
    DOI: 10.1007/978-3-540-37007-9_12
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