Section 5 is devoted to revealing the closed relation between weakly approximate solutions of vector optimization and variational inequality involving set-valued cone-subinvex mappings.
In passing, it is worth noting that some stability results are available for the vector optimization and vector equilibrium problems with a sequence of sets converging in the sense of Painleve-Kuratowski (see [11 13]).
It is well known that the vector equilibrium problem (VEP, in short) is a very general mathematical model, which embraces the formats of several disciplines, as those for Nash equilibria, those from Game Theory, those from (Vector) Optimization and (Vector) Variational Inequalities, and so on (see [1 4]).
In [1 7] and the references therein, the authors have studied the existence and the stability of (vector-valued, set-valued, semi-infinite vector) optimization and multiobjective optimization problems, while the author in [2] shows that most of the weakly efficient solution sets of multiobjective optimization problems (in the sense of Baire category) are stable.
Along with the development of vector optimization and motivated by the wide usefulness of EVP, many authors have been interested in obtaining this principle for vector-valued functions and set-valued mappings; see [3, 5, 8, 9, 11 15, 21, 22, 25].
Er, Yumusak and Temurtas [ 18] presented a comparative chest diagnosis; for chronic obstructive pulmonary, pneumonia, asthma, tuberculosis and lung cancer diseases which was realized by using multilayer, probabilistic, learning vector optimization, and generalized regression.
We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings.
Meanwhile, we obtain the optimality conditions for vector optimization problems and vector variational inequality problems with constraints.
As interesting applications of the results of Section 3, we obtain the optimality conditions for vector optimization problems and vector variational inequality problems.
Mond and Weir [9] presented two pairs of symmetric dual vector optimization problems and obtained symmetric duality results concerning pseudoconvex and pseudoconcave functions.
For a bifunction, the vector equilibrium problem (for short, VEP) is to find such that (1.2). which is a unified model of several known problems, for instance, vector variational and variational-like inequality problems, vector complementarity problem, vector optimization problem, and vector saddle point problem; see, for example, [3, 8, 17, 18] and references therein.
