Variational inequality

Operator (biology)

Lipschitz continuity

Mathematics

Economic growth

Line search

Transcription factor

Gene

Algorithm

Mathematical optimization

Mathematical analysis

Political science

RADIUS

Norm (philosophy)

Repressor

Economics

Chemistry

Convergence (economics)

Computer science

Fixed point

Applied mathematics

Computer security

Biochemistry

Projection (relational algebra)

Hilbert space

Computational and Applied Mathematics

In this paper, we propose a new extragradient method consisting of the hybrid steepest descent method, a single projection method and an Armijo line searching the technique for approximating a solution of variational inequality problem and finding the fixed point of demicontractive mapping in a real Hilbert space. The essence of this algorithm is that a single projection is required in each iteration and the step size for the next iterate is determined in such a way that there is no need for a prior estimate of the Lipschitz constant of the underlying operator. We state and prove a strong convergence theorem for approximating common solutions of variational inequality and fixed points problem under some mild conditions on the control sequences. By casting the problem into an equivalent problem in a suitable product space, we are able to present a simultaneous algorithm for solving the split equality problem without prior knowledge of the operator norm. Finally, we give some numerical examples to show the efficiency of our algorithm over some other algorithms in the literature.

A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem