Bài viết tổng quát cho sự bất bình đẳng Variational với một công thức thiết lập giá trị một số kết quả vô hướng và bất bình đẳng Minty vector biến phân loại khác biệt. Nó khẳng định rằng sự tồn tại của một giải pháp của bất đẳng thức biến phân (thiết lập giá trị) là tương đương với một tài sản ngày càng tăng cùng-quang chức năng thiết lập có giá trị. | Vietnam Journal of Mathematics 35 1 2007 81-106 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2007 Some Remarks on Set-Valued Minty Variational Inequalities Giovanni P. Crespi1 Ivan Ginchev2 and Matteo Rocca3 1 Universite de la Vallée d Aoste Faculty of Economics Aosta Italy 2 Technical University of Varna Department of Mathematics Varna Bulgaria University of Insubria Department of Economics 21100 Varese Italy 3 University of Insubria Department of Economics Varese Italy Received July 21 2006 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. 2000 Mathematics Subject Classification 49J40 49J52 49J53 90C29 47J20. Keywords Minty variational inequalities vector variational inequalities set-valued optimization increasing-along-rays property generalized .
