Minimum principle sufficiency for a Variational inequality with Pseudomonotone mapping

For a variational inequality problem (VIP) with a psudomonotone mapping F on its solution set C∗, we give equivalent statements for C∗ to be determined by the zeroes γ(C∗) of the primal gap function of VIP, where C∗ 2 C∗. One sufficient condition is also presented in terms of weaker sharpness of C∗. With the psudomonotonicityλ of F on C∗ being characterized, C∗ turns out to coincide with the zeroes λ(C∗) of the dual gap function of VIP. If also F has the same direction on γ(C∗), then γ(C∗) coincides with C∗, λ(C∗), and the solution set C∗ of the dual variational inequality problem. This has further been shown to be equivalent to saying that F is constant on γ(C∗) when F is psudomonotonone+ on C∗.