G. Van Der Laan et al., Existence and approximation of robust solutions of variational inequality problems over polytopes, SIAM J CON, 37(2), 1998, pp. 333-352
We study nonlinear variational inequality problems over polytopes from a vi
ewpoint of stability and propose a new solution concept. Extending an earli
er 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 p
oint. 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 continuou
s function on a polytope. The algorithm can be briefly stated as follows. S
tarting with any point in the relative interior of a polytope, the algorith
m generates a piecewise linear path which leads to an approximate robust st
ationary point of any a priori chosen accuracy within a finite number of st
eps. Moreover, we also discuss several numerical examples and apply the new
concept to noncooperative games and economic equilibrium problems.