A Projection Algorithm for Non-Monotone Variational Inequalities

Abstract

We introduce a projection-type algorithm for solving the variational inequality problem for point-to-set operators, and study its convergence properties. No monotonicity assumption is used in our analysis. The operator defining the problem is only assumed to be continuous in the point-to-set sense, i.e., inner-and outer-semicontinuous. Additionally, we assume non-emptiness of the so-called dual solution set. We prove that the whole sequence of iterates converges to a solution of the variational inequality. Moreover, we provide numerical experiments illustrating the behaviour of our iterates.Through several examples, we provide a comparison with a recent similar algorithm.

An inertial Tseng's extragradient method for solving multi-valued variational inequalities with one projection

“…We proved the convergence of the sequences generated by the proposed algorithm and presented some numerical experiments to illustrate the efficiency of our method. Compared with those algorithms in [6,16], only one projection is needed at each iterate in our method. Our method is also different from that in [11].…”

“…Further, Fang and Chen [12] extended the subgradient extragradient algorithm in [7] to solve multi-valued variational inequality(1.1). Recently, Burachik and Milln [6] suggesteded a projection-type algorithm for solving (1.1), in which the next iterate is a projection of the initial point onto the intersection of some suitable convex subsets. He et al [16] proposed two projection-type algorithms for solving the multivalued variational inequality and studied the convergence of the proposed algorithms.…”

Stochastic Relaxed Inertial Forward-Backward-Forward splitting for Monotone Inclusions in Hilbert spaces

“…Since F is upper semi-continuous and monotone, Minty solutions coincide with Stampacchia solutions, implying that there exists x * ∈ F( x) such that x * , y − x ≥ 0 for all y ∈ C (see e.g. [18]). Consider now the gap program…”

An algorithm for best generalised rational approximation of continuous functions

“…In [7], the authors introduce an algorithm for solving variational inequalities, when the operator is pseudo-convex, subject to some continuity requirements. In the following example we show that the subdifferential of the function Ψ f is not necessarily inner semi-continuous, and therefore does not satisfy the requirements from [7].…”