Unconstrained Optimization Reformulations of Variational Inequality Problems

Error Bounds of Generalized D-Gap Functions for Nonsmooth and Nonmonotone Variational Inequality Problems

“…Peng and Fukushima [25] gave a global error bound result for D-gap functions under the assumption that F is smooth, (globally) Lipschitz on R n and strongly monotone. Their result was extended by Yamashita, Taji and Fukushima [33] to cover generalized D-gap functions. Recently, Huang and Ng [13] gave an example showing that the D-gap functions might not provide global error bounds if the globally Lipschitz assumption of F is dropped.…”

“…Among these merit functions, D-gap functions and generalized D-gap functions for (VI) problems are particularly interesting, because they cast (VI) problems as equivalent unconstrained optimization problems (cf. [24,33]). Peng and Fukushima [25] gave a global error bound result for D-gap functions under the assumption that F is smooth, (globally) Lipschitz on R n and strongly monotone.…”

“…[33] replaced the function 1 2 x−y 2 in (3.1) by a more general function ϕ and thereby introduced generalized D-gap functions θ ϕ ab and generalized regularized gap functions θ ϕ c respectively defined by (for 0 < a < b and c > 0)…”

“…Here ϕ is a real-valued function on R n × R n satisfying the following properties (P6 and P7 are redundant; see [13,33]): P1: ϕ is continuously differentiable on R n × R n . P2: ϕ(x, y) ≥ 0 for all x, y ∈ R n and the equality holds if and only if x = y. P3: {ϕ(x, ·) : x ∈ R n } is uniformly strongly convex in the sense that there exists a positive real number β such that for all x ∈ R n ,…”

“…We always assume that a, b, c > 0 and a < b. The following proposition collects some useful facts for the generalized D-gap functions in which part (i) − (iii) are taken from [31], part (iv) − (vii) from [33] and part (viii) from [19]. (i) For every x ∈ R n , there exists a unique vector y ϕ c (x) ∈ P at which the supremum in (3.3) is attained, i.e.,…”