Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems
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-p...
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| Format: | Article |
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MDPI AG
2023-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/9/2102 |
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| author | Eltiyeb Ali Salem Mahdi |
| author_facet | Eltiyeb Ali Salem Mahdi |
| author_sort | Eltiyeb Ali |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T04:13:11Z |
| format | Article |
| id | doaj.art-69662295787042f0990a7ea0bd35a0ac |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:13:11Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-69662295787042f0990a7ea0bd35a0ac2023-11-17T23:20:01ZengMDPI AGMathematics2227-73902023-04-01119210210.3390/math11092102Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear SystemsEltiyeb Ali0Salem Mahdi1Department of Mathematics, College of Science and Arts—Sharourah, Najran University, P.O. Box 1988, Najran 68341, Saudi ArabiaDepartment of Mathematics & Computer Science, Faculty of Science, Alexandria University, Alexandria 5424041, EgyptIn 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.https://www.mdpi.com/2227-7390/11/9/2102conjugate gradient methodsperformance profilesbacktracking line searchnumerical comparisons and evaluation criteria |
| spellingShingle | Eltiyeb Ali Salem Mahdi Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems Mathematics conjugate gradient methods performance profiles backtracking line search numerical comparisons and evaluation criteria |
| title | Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems |
| title_full | Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems |
| title_fullStr | Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems |
| title_full_unstemmed | Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems |
| title_short | Adaptive Hybrid Mixed Two-Point Step Size Gradient Algorithm for Solving Non-Linear Systems |
| title_sort | adaptive hybrid mixed two point step size gradient algorithm for solving non linear systems |
| topic | conjugate gradient methods performance profiles backtracking line search numerical comparisons and evaluation criteria |
| url | https://www.mdpi.com/2227-7390/11/9/2102 |
| work_keys_str_mv | AT eltiyebali adaptivehybridmixedtwopointstepsizegradientalgorithmforsolvingnonlinearsystems AT salemmahdi adaptivehybridmixedtwopointstepsizegradientalgorithmforsolvingnonlinearsystems |