
Vietnam Journal of Mathematics 35:1(2007)
81106

Some Remarks on SetValued Minty Variational
Inequalities

Giovanni P. Crespi,
Ivan Ginchev, and Matteo Rocca

Abstract. The paper generalizes to
variational inequalities with a setvalued formulation some results for
scalar and vector Minty variational inequalities of differential type. It
states that the existence of a solution of the (setvalued) variational
inequality is equivalent to an increasingalongrays property of the
setvalued function and implies that the solution is also a point of
efficiency (minimizer) for the underlying setvalued optimization problem.
A special approach is proposed in order to treat in a uniform way the cases
of several efficient points. Applications to aminimizers (absolute or ideal efficient points) and wminimizers (weakly efficient
points) are given. A comparison among the commonly accepted notions of
optimality in setvalued optimization and these which appear to be related
with the setvalued variational inequality leads to two concepts of
minimizers,called here point minimizers and set minimizers. Further the
role of generalized (quasi)convexity is highlighted in the process of
defining a class of functions, such that each solution of the setvalued
optimization problem solves also the setvalued variational inequality. For
aminimizers and wminimizers it appears to be useful
*quasiconvexity and Cquasiconvexity
for setvalued functions.


2000 Mathematics Subject Classification: 49J40, 49J52,
49J53, 90C29, 47J20.

Keywords: Minty variational inequalities, vector
variational inequalities, setvalued optimization, increasingalongrays
property, generalized quasiconvexity.


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by Vietnam Academy of Science and Technology & Vietnam Mathematical
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