Solving Mixed Variational Inequalities Beyond Convexity

Abstract

We show that Malitsky’s recent Golden Ratio Algorithm for solving convex mixed variational inequalities can be employed in a certain nonconvex framework as well, making it probably the first iterative method in the literature for solving generalized convex mixed variational inequalities, and illustrate this result by numerical experiments.

Relaxed-inertial proximal point type algorithms for quasiconvex minimization

“…The proximity operator on K of parameter β > 0 of h at x ∈ R n is defined as Prox βh : R n ⇒ R n where ). Although in most of the literature the proximity operator of a function is considered (in the spirit of "full splitting") on the whole space, there are works like [10,16,17] where the employed functions are not split from the corresponding sets, as defined above.…”

Inertial Forward-Backward-Forward Method for Non-convex Mixed Variational Inequalities

“…In this section, we provide computational experiments to compare our proposed Algorithm 2 to the existing state-of-the-art Algorithm 1 (proposed in [17]) using test examples below. All codes were written in MATLAB R2020b and performed on a PC Desktop Intel(R) Core(TM) i7-6600U CPU @ 3.00GHz 3.00 GHz, RAM 32.00 GB.…”

“…As explained in [17], the cost functions ϕ 1 , ϕ 3 and ϕ 5 are prox-convex with constant α for any α > 0, while ϕ 2 and ϕ 4 convex. More details on these cost functions can be found in [1,2,10,17,20,31].…”

“…However when h in MVI ( 1) is non-convex function, then MVI (1) becomes harder to solve. This is because the solution set may be empty even when K is compact and convex set (see [30,Example 3.1], [22, page 127] and [17] for more details).…”

“…Grad and Lara [17] proved that the sequences {x n } and {z n } generated by Algorithm 1 converge to a solution of non-convex MVI (1). As far as we know, the Algorithm 1 proposed in [17] is the first iterative method in the literature to solve non-convex MVI (1).…”