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Vietnam Journal of Mathematics 35:1(2007)
81-106
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Some Remarks on Set-Valued Minty Variational
Inequalities
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Giovanni P. Crespi,
Ivan Ginchev, and Matteo Rocca
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Abstract. The paper generalizes to
variational inequalities with a set-valued formulation some results for
scalar and vector Minty variational inequalities of differential type. It
states that the existence of a solution of the (set-valued) variational
inequality is equivalent to an increasing-along-rays property of the
set-valued function and implies that the solution is also a point of
efficiency (minimizer) for the underlying set-valued optimization problem.
A special approach is proposed in order to treat in a uniform way the cases
of several efficient points. Applications to a-minimizers (absolute or ideal efficient points) and w-minimizers (weakly efficient
points) are given. A comparison among the commonly accepted notions of
optimality in set-valued optimization and these which appear to be related
with the set-valued 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 set-valued
optimization problem solves also the set-valued variational inequality. For
a-minimizers and w-minimizers it appears to be useful
*-quasiconvexity and C-quasiconvexity
for set-valued functions.
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2000 Mathematics Subject Classification: 49J40, 49J52,
49J53, 90C29, 47J20.
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Keywords: Minty variational inequalities, vector
variational inequalities, set-valued optimization, increasing-along-rays
property, generalized quasiconvexity.
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Established
by Vietnam Academy of Science and Technology & Vietnam Mathematical
Society
Published
by Springer since January 2013
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