INERTIAL RELAXED TSENG METHOD FOR SOLVING VARIATIONAL INEQUALITY PROBLEM IN HILBERT SPACE

The research efforts of this paper is to present a new inertial relaxed Tseng extrapolation method with weaker conditions for approximating the solution of a variational inequality problem, where the underlying operator is only required to be pseudomonotone. The strongly pseudomonotonicity and inverse strongly monotonicity assumptions which the existing literature used are successfully weakened. The strong convergence of the proposed method to a minimum-norm solution of a variational inequality problem are established. Furthermore, we present an application and some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature.

Greedy Randomized and Maximal Weighted Residual Kaczmarz Methods with Oblique Projection

For solving large-scale consistent linear system, a greedy randomized Kaczmarz method with oblique projection and a maximal weighted residual Kaczmarz method with oblique projection are proposed. By using oblique projection, these two methods greatly reduce the number of iteration steps and running time to find the minimum norm solution, especially when the rows of matrix A are close to linear correlation. Theoretical proof and numerical results show that the greedy randomized Kaczmarz method with oblique projection and the maximal weighted residual Kaczmarz method with oblique projection are more effective than the greedy randomized Kaczmarz method and the maximal weighted residual Kaczmarz method respectively.

On the minimum-norm solution of convex quadratic programming

We discuss some basic concepts and present a numerical procedure for finding the minimum-norm solution of convex quadratic programs (QPs) subject to linear equality and inequality constraints. Our approach is based on a theorem of alternatives and on a convenient characterization of the solution set of convex QPs. We show that this problem can be reduced to a simple constrained minimization problem with a once-differentiable convex objective function. We use finite termination of an appropriate Newton's method to solve this problem. Numerical results show that the proposed method is efficient.

Iterative Algorithms for Split Common Fixed Point Problem Involved in Pseudo-Contractive Operators without Lipschitz Assumption

Two iterative algorithms are suggested for approximating a solution of the split common fixed point problem involved in pseudo-contractive operators without Lipschitz assumption. We prove that the sequence generated by the first algorithm converges weakly to a solution of the split common fixed point problem and the second one converges strongly. Moreover, the sequence { x n } generated by Algorithm 3 strongly converges to z = proj S 0 , which is the minimum-norm solution of problem (1). Numerical examples are included.