A note on Minty type vector variational inequalities

¹ Université de la Vallé d'Aoste, Faculty of Economics, 11100 Aosta, Italy; g.crespi@univda.it
² Technical University of Varna, Department of Mathematics, 9010 Varna, Bulgaria; ginchev@ms3.tu-varna.acad.bg
³ University of Insubria, Department of Economics, 21100 Varese; Italy; mrocca@eco.uninsubria.it

Received: 10 November 2004
Accepted: 6 January 2006

Abstract

The existence of solutions to a scalar Minty variational inequality of differential type is usually related to monotonicity property of the primitive function. On the other hand, solutions of the variational inequality are global minimizers for the primitive function. The present paper generalizes these results to vector variational inequalities putting the Increasing Along Rays (IAR) property into the center of the discussion. To achieve that infinite elements in the image space Y are introduced. Under quasiconvexity assumptions we show that solutions of generalized variational inequality and of the primitive optimization problem are equivalent. Finally, we discuss the possibility to generalize the scheme of this paper to get further applications in vector optimization.