Mixed polynomial variational inequalities

The goal of this paper is to introduce a class of mixed polynomial variational inequalities, which is a natural generalization of the affine variational inequality and the tensor variational inequality, and a special case of the mixed variational inequality. It is shown that a class of polynomial optimization problem and a class of m -person noncooperative game can be reformulated as a mixed polynomial variational inequality. Firstly, some classes of structured tensor tuples are introduced and the relationship between them is discussed. Then, a new asymptotic function (denoted by m -asymptotic function) is introduced and some basic properties are investigated. An equivalent characterization for the nonexistence of solutions is given by using the exceptional family of elements. Finally, the nonemptiness and compactness of the solution sets of the mixed polynomial variational inequalities with some special structured tensors and m -asymptotic function are proved and then the uniqueness of the solution is further investigated.