A remark on vector-valued equilibria and generalized monotonicity. (English) Zbl 0914.90235

Summary: It is the purpose of this note to show that existence results for vector-valued equilibria, of the type considered recently by Q. H. Ansari [‘Vector equilibrium problems and vector variational inequalities’, presented at the 2nd World Congress of Nonlinear Analysts, Althens 1996] and M. Bianchi, N. Hadjisavvas and S. Schaible [J. Optimization Theory Appl. 92, No. 3, 527-542 (1997; Zbl 0878.49007)], can be deduced in a straightforward way from corresponding results about scalar-valued equilibria. We shall proceed as follows. First we prove an existence result for scalar-valued equilibria, employing a certain notion of generalized monotonicity, and from this we deduce several results for the vector-valued case, by using appropriate gauge functions.


90C29 Multi-objective and goal programming


Zbl 0878.49007