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∗ ∈ C∗. One sufficient condition is also presented in terms of weaker sharpness of C∗. With the psudomonotonicity∗ of F onC∗ 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∗. Key–Words: Variational inequality, minimum principle sufficiency, weaker sharpness, pseudomonotonicity∗, gap functions