Properties of gap functions for mixed variational inequalities

Various gap functions for a class of mixed variational inequalities containing a $P$-mapping function and a convex separable function which are not necessarily differentiable are considered. Such problems have a number of applications in mathematical physics, economics, and operations research. The initial problem is shown to be equivalent to a constrained optimization problem for a conventional gap function which may be non-differentiable. At the same time, the $D$-gap function allows one to reduce the initial problem to the problem of finding stationary points of a continuously differentiable function. This latter problem can be solved by standard unconstrained differentiable optimization methods.