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

Eltiyeb Ali and Salem Mahdi ()
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Eltiyeb Ali: Department of Mathematics, College of Science and Arts—Sharourah, Najran University, P.O. Box 1988, Najran 68341, Saudi Arabia
Salem Mahdi: Department of Mathematics & Computer Science, Faculty of Science, Alexandria University, Alexandria 5424041, Egypt

Mathematics, 2023, vol. 11, issue 9, 1-35

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.

Keywords: conjugate gradient methods; performance profiles; backtracking line search; numerical comparisons and evaluation criteria (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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