Abstract
A mixed monotone variational inequality (MMVI) problem in a Hilbert space is formulated to find a point such that for all , where is a monotone operator and is a proper, convex, and lower semicontinuous function on . Iterative algorithms are usually applied to find a solution of an MMVI problem. We show that the iterative algorithm introduced in the work of Wang et al., (2001) has in general weak convergence in an infinite-dimensional space, and the algorithm introduced in the paper of Noor (2001) fails in general to converge to a solution.
Highlights
Let H be a real Hilbert space with inner product ·, · and norm ·, and let T be an operator with domain D T and range R T in H
Recall that T is monotone if its graph G T : { x, y ∈ H × H : x ∈ D T, y ∈ T x} is a monotone set in H × H. This means that T is monotone if and only if x, y, x, y ∈ G T ⇒ x − x, y − y ≥ 0
It is known that T is monotone if and only of for each ρ > 0, the resolvent JρT is nonexpansive, and T is maximal monotone if and only of for each ρ > 0, the resolvent JρT is nonexpansive and defined on the entire space H
Summary
Let H be a real Hilbert space with inner product ·, · and norm · , and let T be an operator with domain D T and range R T in H. ∂φ x : z ∈ H : φ y ≥ φ x y − x, z , ∀y ∈ H It is well known cf 1 that ∂φ is a maximal monotone operator. The mixed monotone variational inequality (MMVI) problem is to find a point u∗ ∈ H with the property. T u∗, v − u∗ φ v − φ u∗ ≥ 0, ∀v ∈ H, 1.3 where T is a monotone operator and φ is a proper, convex, and lower semicontinuous function on H. It is known that T is monotone if and only of for each ρ > 0, the resolvent JρT is nonexpansive, and T is maximal monotone if and only of for each ρ > 0, the resolvent JρT is nonexpansive and defined on the entire space H. If T is a single-valued, strongly monotone i.e., T x − T y, x − y ≥ τ x − y 2 for all x, y ∈ K and some τ > 0 , and Lipschitzian i.e., T x − T y ≤ L x − y for some L > 0 and all x, y ∈ D T operator on K, the sequence {xk} generated by the iterative algorithm xk 1 PK I − ρT xk, k ≥ 0, 1.7 where I is the identity operator and PK is the metric projection of H onto K, and the initial guess x0 ∈ H is chosen arbitrarily, converges strongly to the unique solution of VI 1.5 provided, ρ > 0 is small enough
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