Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems

Abstract

In this paper, a two-point step-size gradient technique is proposed by which the approximate solutions of a non-linear system are found. The two-point step-size includes two types of parameters deterministic and random. A new adaptive backtracking line search is presented and combined with the two-point step-size gradient to make it globally convergent. The idea of the suggested method depends on imitating the forward difference method by using one point to estimate the values of the gradient vector per iteration where the number of the function evaluation is at most one for each iteration. The global convergence analysis of the proposed method is established under actual and limited conditions. The performance of the proposed method is examined by solving a set of non-linear systems containing high dimensions. The results of the proposed method is compared to the results of a derivative-free three-term conjugate gradient CG method that solves the same test problems. Fair, popular, and sensible evaluation criteria are used for comparisons. The numerical results show that the proposed method has merit and is competitive in all cases and superior in terms of efficiency, reliability, and effectiveness in finding the approximate solution of the non-linear systems.

A Family of Developed Hybrid Four-Term Conjugate Gradient Algorithms for Unconstrained Optimization with Applications in Image Restoration

“…Accordingly, many authors have suggested several modificationsof the classical methods that deal with optimization problems, e.g, the Newton method [3], the quasi-Newton method [4], the semi-gradient method [5], the hybrid gradient meta-heuristic algorithm [6,7], and the CG-method [8][9][10][11].…”