Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator

Abstract

Two inertial subgradient extragradient algorithms for solving variational inequality problems involving pseudomonotone operator are proposed in this article. The iterative schemes use self-adaptive step sizes which do not require the prior knowledge of the Lipschitz constant of the underlying operator. Furthermore, under mild assumptions, we show the weak and strong convergence of the sequences generated by the proposed algorithms. The strong convergence in the second algorithm follows from the use of viscosity method. Numerical experiments both in finite- and infinite-dimensional spaces are reported to illustrate the inertial effect and the computational performance of the proposed algorithms in comparison with the existing state of the art algorithms.

Convergence Analysis of Self-Adaptive Inertial Extra-Gradient Method for Solving a Family of Pseudomonotone Equilibrium Problems with Application

“…where y ∈ H n . It follows from (30), (34), (35), and the boundedness of {u n } that the right hand side tends to zero. Due to λ n k > 0, condition (f3), and v n k z, we have…”

“…The construction of new iterative schemes and the modification of existing methods, as well as the study their convergence analysis, constitute an important research direction in equilibrium problem theory. Several methods have been developed in the past few years to approximate the solution of an equilibrium problem in finite and infinite dimensional real Hilbert spaces, i.e., extragradient methods [7][8][9][10][11][12][13][14][15][16], subgradient methods [17][18][19][20][21][22], inertial methods [23][24][25] and methods for particular classes of equilibrium problems [26][27][28][29][30][31][32][33][34][35].…”

Projection method with inertial step for nonlinear equations: Application to signal recovery

“…Polyak [54] introduced an inertial extrapolation as an acceleration mechanism to solve the smooth convex minimization problem based on the heavy ball methods of the two-order time dynamical system. The inertial algorithm is a two-step iterative method that uses the previous two iterates to compute the next iterate, and it can be thought of as a procedure of speeding up the convergence properties, see [54,17,14,5,15]. An abundant literature has also been devoted on this subject with a lot of fast iterative algorithms constructed using the inertial extrapolation, for instance, inertial forward-backward splitting methods [18,50], inertial Mann method [58], inertial Douglas-Rachford splitting method [24], inertial ADMM [25], inertial subgradient extragradient method [59], inertial forward-backward-forward method [22], inertial proximal-extragradient method [23], and inertial contraction method [32].…”

A Weak Convergence Self-Adaptive Method for Solving Pseudomonotone Equilibrium Problems in a Real Hilbert Space

“…Many methods have been well-established to figure out the solution of an equilibrium problem (1) in finite and infinite-dimensional spaces. Some of these algorithms involve projection methods [5][6][7][8], the proximal point methods [9,10], the extragradient methods with or without line searches [11][12][13][14][15][16][17][18], the methods using the inertial effect [19][20][21][22] and other methods in [23][24][25][26][27][28].…”