We study nonlinear variational inequality problems over polytopes from a viewpoint of stability and propose a new solution concept. Extending an earlier concept proposed by Yang [Z. Yang, SIAM J. Control Optim., 34 (1996), pp. 491-506] on the unit simplex, we will introduce the concept of the robust stationary point, which is a refinement of the concept of the stationary point. Though a stationary point need not be robust, it is shown that every continuous function on a polytope has a robust stationary point. We develop a simplicial algorithm to compute a robust stationary point of a continuous function on a polytope. The algorithm can be briefly stated as follows. Starting with any point in the relative interior of a polytope, the algorithm generates a piecewise linear path which leads to an approximate robust stationary point of any a priori chosen accuracy within a finite number of steps. Moreover, we also discuss several numerical examples and apply the new concept to noncooperative games and economic equilibrium problems.