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Some remarks on the Minty vector variational principle. (English) Zbl 1152.49007
Summary: In scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Dordrecht: Kluwer Academic Publishers. Appl. Optim. 13, 93–99 (1998; Zbl 0909.90253)] and subsequently in [X. M. Yang, X. Q. Yang and K. L. Teo, J. Optim. Theory Appl. 121, No. 1, 193–201 (2004; Zbl 1140.90492)]. In these papers, in the particular case of a differentiable objective function $f$ taking values in ${ℝ}^{m}$ and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions.
##### MSC:
 49J40 Variational methods including variational inequalities 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions)