A multi step inertial algorithm for approximating a common solution of split generalized mixed equilibrium and minimization problems

A Totally Relaxed Self-Adaptive Subgradient Extragradient Scheme for Equilibrium and Fixed Point Problems in a Banach Space

“…Let f : C × C → R ∪ {+∞} be a bifunction such that C ⊂ int(dom(f, •)), then for every x ∈ C, the Equilibrium Problem (EP) (see [3,14]), is to find a point x * ∈ C such that f (x * , y) ≥ 0, for all y ∈ C. (1.1) The EP is a generalization of many important optimization problems, such as Variational Inequality Problem (VIP), Fixed Point Problem (FPP) and so on (see [6,14] and the references therein). In particular, if f (x, y) = ⟨Ax, y − x⟩, where A : C → E * , is a nonlinear mapping, then EP (C, f ) (1.1) reduces to the classical VIP introduced by Stampacchia [47] (see also [36,38,41,52]), which is to find a point x * ∈ C such that ⟨Ax * , y − x * ⟩ ≥ 0, for all y ∈ C. (1.2) There are two important directions of research on EP: These are the existence of solution of EP and other related problems (see [14,29] for more details) and the development of iterative algorithms for approximating the solution of EP, its several generalizations and related optimization problems (see [1,12,13,33,34,[42][43][44] and the references therein).…”

A self-adaptive algorithm with multi-step inertia for solving convex bilevel optimization problems

“…The inertial step usually has good acceleration, therefore it is widely applied for solving many kinds of problems, such as variational inequality, generalized mixed equilibrium, ect., see for instance [5,23,20].…”