In this paper, we introduce hybrid inertial subgradient extragradient algorithms with line-search process for solving a variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, we prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

Let X be a uniformly convex and q-uniformly smooth Banach space with $1< q\leq 2$1<q≤2 . In the framework of this space, we are concerned with a composite gradient-like implicit rule for solving a hierarchical monotone variational inequality with the constraints of a system of monotone variational inequalities, a variational inclusion and a common fixed point problem of a countable family of nonlinear operators $\{S_{n}\}^{\infty }_{n=0}${Sn}n=0∞ . Our rule is based on the Korpelevich extragradient method, the perturbation mapping, and the W-mappings constructed by $\{S_{n}\}^{\infty }_{n=0}${Sn}n=0∞ .

In this paper, we propose viscosity algorithms with two different inertia parameters for solving fixed points of nonexpansive and strictly pseudocontractive mappings. Strong convergence theorems are obtained in Hilbert spaces and the applications to the signal processing are considered. Moreover, some numerical experiments of proposed algorithms and comparisons with existing algorithms are given to the demonstration of the efficiency of the proposed algorithms. The numerical results show that our algorithms are superior to some related algorithms.

In this paper, we study the sparsest solutions of linear complementarity problems (LCPs), which study has many applications, such as bimatrix games and portfolio selections. Mathematically, the underlying model is NP-hard in general. By transforming the complementarity constraints into a fixed point equation with projection type, we propose an l1 regularization projection minimization model for relaxation. Through developing a thresholding representation of solutions for a key subproblem of this regularization model, we design a shrinkage-thresholding projection (STP) algorithm to solve this model and also analyze convergence of STP algorithm. Numerical results demonstrate that the STP method can efficiently solve this regularized model and get a sparsest solution of LCP with high quality.

In this paper, we aim to find sparse solutions of co-coercive nonlinear complementarity problems (NCPs). Mathematically, the underlying model is NP-hard in general. Thus an ℓ 1 regularized projection minimization model is proposed for relaxation and an extragradient thresholding algorithm (ETA) is then designed for this regularized model. Furthermore, we analyze the convergence of this algorithm and show any cluster point of the sequence generated by ETA is a solution of NCP. Numerical results demonstrate that the ETA can effectively solve the ℓ 1 regularized model and output very sparse solutions of co-coercive NCPs with high quality.