Research Article
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Year 2021, Volume: 2 Issue: 2, 108 - 119, 15.12.2021

Mezher M. Abed1 Ufuk Öztürk Hisham Mohammed

Abstract

References

  • L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning-III,” Inf. Sci. (Ny)., vol. 9, no. 1, pp. 43–80, 1975, doi: 10.1016/0020-0255(75)90017-1.
  • R. Badard, “The law of large numbers for fuzzy processes and the estimation problem,” Inf. Sci. (Ny)., vol. 28, no. 3, 1982, doi: 10.1016/0020-0255(82)90046-9.
  • P. Diamond, “Fuzzy least squares,” Inf. Sci. (Ny)., vol. 46, no. 3, pp. 141–157, 1988.
  • M. Friedman, M. Ming, and A. Kandel, “Fuzzy linear systems,” Fuzzy sets Syst., vol. 96, no. 2, pp. 201–209, 1998.
  • J. J. Buckley and Y. Qu, “Solving linear and quadratic fuzzy equations,” Fuzzy Sets Syst., vol. 38, no. 1, pp. 43–59, 1990, doi: 10.1016/0165-0114(90)90099-R.
  • J. J. Buckley and Y. Qu, “On using α-cuts to evaluate fuzzy equations,” Fuzzy Sets and Systems, vol. 43, no. 1. Elsevier, p. 125, 1991, doi: 10.1016/0165-0114(91)90026-M.
  • J. J. Buckley and Y. Qu, “Solving fuzzy equations: A new solution concept,” Fuzzy Sets Syst., vol. 39, no. 3, pp. 291–301, 1991, doi: 10.1016/0165-0114(91)90099-C.
  • J. J. Buckley and Y. Qu, “Solving systems of linear fuzzy equations,” Fuzzy Sets Syst., vol. 43, no. 1, pp. 33–43, 1991, doi: 10.1016/0165-0114(91)90019-M.
  • S. Abbasbandy and B. Asady, “Newton’s method for solving fuzzy nonlinear equations,” Appl. Math. Comput., vol. 159, no. 2, pp. 349–356, Jul. 2004, doi: 10.1016/j.amc.2003.10.048.
  • M. Mamat, A. Ramli, and M. L. Abdullah, “Broyden’s method for solving fuzzy nonlinear equations,” Adv. Fuzzy Syst., 2010, doi: 10.1155/2010/763270.
  • S. Abbasbandy and A. Jafarian, “Steepest descent method for solving fuzzy nonlinear equations,” Appl. Math. Comput., vol. 174, no. 1, pp. 669–675, Jul. 2006, doi: 10.1016/j.amc.2005.04.092.
  • H. M. Khudhur and K. K. Abbo, “A New Type of Conjugate Gradient Technique for Solving Fuzzy Nonlinear Algebraic Equations,” in Journal of Physics: Conference Series, 2021, vol. 1879, no. 2, doi: 10.1088/1742-6596/1879/2/022111.
  • H. M. Khudhur and K. K. Abbo, “New hybrid of Conjugate Gradient Technique for Solving Fuzzy Nonlinear Equations,” J. Soft Comput. Artif. Intell., vol. 2, no. 1, pp. 1–8, 2021.
  • M. M. Abed, U. Öztürk, and H. Khudhur, “Spectral CG Algorithm for Solving Fuzzy Non-linear Equations,” Iraqi J. Comput. Sci. Math., vol. 3, no. 1, 2022.
  • L. A. Zadeh, “Fuzzy sets,” Inf. Control, vol. 8, no. 3, pp. 338–353, 1965.
  • D. J. Dubois, Fuzzy sets and systems: theory and applications, vol. 144. Academic press, 1980.
  • H. Zimmermann, “Fuzzy set theory and its applications,” Int. Ser. Manag., 1991.
  • B. Hassan, H. Jabbar, and A. Al-Bayati, “A new class of nonlinear conjugate gradient method for solving unconstrained minimization problems,” 2019, doi: 10.1109/ICCISTA.2019.8830657.
  • B. A. Hassan, “A new type of quasi-newton updating formulas based on the new quasi-newton equation,” Numer. Algebr. Control Optim., vol. 10, no. 2, pp. 227–235, 2020, doi: 10.3934/naco.2019049.
  • WOLFE P, “CONVERGENCE CONDITIONS FOR ASCENT METHODS,” SIAM Rev., vol. 11, no. 2, pp. 226–235, 1969, doi: 10.1137/1011036.
  • H. M. Azzam, H. N. Jabbar, and K. K. Abo, “Four–Term Conjugate Gradient (CG) Method Based on Pure Conjugacy Condition for Unconstrained Optimization,” Kirkuk Univ. Journal-Scientific Stud., vol. 13, no. 2, pp. 101–113, 2018, doi: 10.32894/kujss.2018.145720.
  • Y. A. Laylani, K. K. Abbo, and H. M. Khudhur, “Training feed forward neural network with modified Fletcher-Reeves method,” J. Multidiscip. Model. Optim., vol. 1, no. 1, pp. 14–22, 2018, [Online]. Available: http://dergipark.gov.tr/jmmo/issue/38716/392124#article_cite.
  • A. S. Ahmed, “Optimization Methods For Learning Artificial Neural Networks,” University of Mosul, 2018.
  • A. S. Ahmed, H. M. Khudhur, and M. S. Najmuldeen, “A new parameter in three-term conjugate gradient algorithms for unconstrained optimization,” Indones. J. Electr. Eng. Comput. Sci., vol. 23, no. 1, 2021, doi: 10.11591/ijeecs.v23.i1.pp338-344.
  • H. M. Khudhur, “Numerical and analytical study of some descent algorithms to solve unconstrained Optimization problems,” University of Mosul, 2015.
  • K. K. Abbo and H. M. Khudhur, “New A hybrid conjugate gradient Fletcher-Reeves and Polak-Ribiere algorithm for unconstrained optimization,” Tikrit J. Pure Sci., vol. 21, no. 1, pp. 124–129, 2015.
  • K. K. Abbo and H. M. Khudhur, “New A hybrid Hestenes-Stiefel and Dai-Yuan conjugate gradient algorithms for unconstrained optimization,” Tikrit J. Pure Sci., vol. 21, no. 1, pp. 118–123, 2015.
  • K. K. ABBO, Y. A. Laylani, and H. M. Khudhur, “A NEW SPECTRAL CONJUGATE GRADIENT ALGORITHM FOR UNCONSTRAINED OPTIMIZATION,” Int. J. Math. Comput. Appl. Res., vol. 8, pp. 1–9, 2018.
  • K. K. Abbo, Y. A. Laylani, and H. M. Khudhur, “Proposed new Scaled conjugate gradient algorithm for Unconstrained Optimization,” Int. J. Enhanc. Res. Sci. Technol. Eng., vol. 5, no. 7, 2016.
  • Z. M. Abdullah, M. Hameed, M. K. Hisham, and M. A. Khaleel, “Modified new conjugate gradient method for Unconstrained Optimization,” Tikrit J. Pure Sci., vol. 24, no. 5, pp. 86–90, 2019.
  • H. M. Khudhur and K. K. Abbo, “A New Conjugate Gradient Method for Learning Fuzzy Neural Networks,” J. Multidiscip. Model. Optim., vol. 3, no. 2, pp. 57–69, 2020.
  • R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J., vol. 7, no. 2, pp. 149–154, 1964, doi: 10.1093/comjnl/7.2.149.
  • C. Witzgall and R. Fletcher, “Practical Methods of Optimization.,” Math. Comput., vol. 53, no. 188, p. 768, 1989, doi: 10.2307/2008742.
  • E. Polak and G. Ribiere, “Note sur la convergence de méthodes de directions conjuguées,” Rev. française d’informatique Rech. opérationnelle. Série rouge, vol. 3, no. 16, pp. 35–43, 1969, doi: 10.1051/m2an/196903r100351.
  • M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. (1934)., vol. 49, no. 6, p. 409, 1952, doi: 10.6028/jres.049.044.
  • Y. H. Dai and Y. Yuan, “A nonlinear conjugate gradient method with a strong global convergence property,” SIAM J. Optim., vol. 10, no. 1, pp. 177–182, 1999, doi: 10.1137/S1052623497318992.
  • Z.-J. Shi and J. Guo, “A new algorithm of nonlinear conjugate gradient method with strong convergence,” Comput. Appl. Math., vol. 27, pp. 93–106, 2008.

Spectral Fletcher (CD) Algorithm for Solving Fuzzy Non-Linear Equations

Year 2021, Volume: 2 Issue: 2, 108 - 119, 15.12.2021

Mezher M. Abed1 Ufuk Öztürk Hisham Mohammed

Abstract

A conjugate gradient method is a powerful tool for solving large-scale miniaturization issues, with applications in arithmetic, chemistry, physics, engineering, medicine, and other fields. In this paper, we introduce a new spectral conjugate gradient algorithm, whose derivation is based on the Fletcher (CD) and Newton algorithms based on the solely coupling condition, which is introduced in this study. The significance of the research is in identifying a suitable algorithm. Because the Buckley and Qu methods are ineffectual in solving all types of ambiguous equations, and the conjugate gradient approach does not require a Hessian matrix (second partial derivatives of functions) in the solution, it is used to solve all types of ambiguous equations. The suggested method's descent property is demonstrated as long as the α_kstep size matches the strong Wolfe conditions. In many cases, numerical findings demonstrate that the novel technique is more efficient in solving nonlinear fuzzy equations than Fletcher (CD) algorithm.

Keywords

algorithms CG Fletcher fuzzy numerical

References

  • L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning-III,” Inf. Sci. (Ny)., vol. 9, no. 1, pp. 43–80, 1975, doi: 10.1016/0020-0255(75)90017-1.
  • R. Badard, “The law of large numbers for fuzzy processes and the estimation problem,” Inf. Sci. (Ny)., vol. 28, no. 3, 1982, doi: 10.1016/0020-0255(82)90046-9.
  • P. Diamond, “Fuzzy least squares,” Inf. Sci. (Ny)., vol. 46, no. 3, pp. 141–157, 1988.
  • M. Friedman, M. Ming, and A. Kandel, “Fuzzy linear systems,” Fuzzy sets Syst., vol. 96, no. 2, pp. 201–209, 1998.
  • J. J. Buckley and Y. Qu, “Solving linear and quadratic fuzzy equations,” Fuzzy Sets Syst., vol. 38, no. 1, pp. 43–59, 1990, doi: 10.1016/0165-0114(90)90099-R.
  • J. J. Buckley and Y. Qu, “On using α-cuts to evaluate fuzzy equations,” Fuzzy Sets and Systems, vol. 43, no. 1. Elsevier, p. 125, 1991, doi: 10.1016/0165-0114(91)90026-M.
  • J. J. Buckley and Y. Qu, “Solving fuzzy equations: A new solution concept,” Fuzzy Sets Syst., vol. 39, no. 3, pp. 291–301, 1991, doi: 10.1016/0165-0114(91)90099-C.
  • J. J. Buckley and Y. Qu, “Solving systems of linear fuzzy equations,” Fuzzy Sets Syst., vol. 43, no. 1, pp. 33–43, 1991, doi: 10.1016/0165-0114(91)90019-M.
  • S. Abbasbandy and B. Asady, “Newton’s method for solving fuzzy nonlinear equations,” Appl. Math. Comput., vol. 159, no. 2, pp. 349–356, Jul. 2004, doi: 10.1016/j.amc.2003.10.048.
  • M. Mamat, A. Ramli, and M. L. Abdullah, “Broyden’s method for solving fuzzy nonlinear equations,” Adv. Fuzzy Syst., 2010, doi: 10.1155/2010/763270.
  • S. Abbasbandy and A. Jafarian, “Steepest descent method for solving fuzzy nonlinear equations,” Appl. Math. Comput., vol. 174, no. 1, pp. 669–675, Jul. 2006, doi: 10.1016/j.amc.2005.04.092.
  • H. M. Khudhur and K. K. Abbo, “A New Type of Conjugate Gradient Technique for Solving Fuzzy Nonlinear Algebraic Equations,” in Journal of Physics: Conference Series, 2021, vol. 1879, no. 2, doi: 10.1088/1742-6596/1879/2/022111.
  • H. M. Khudhur and K. K. Abbo, “New hybrid of Conjugate Gradient Technique for Solving Fuzzy Nonlinear Equations,” J. Soft Comput. Artif. Intell., vol. 2, no. 1, pp. 1–8, 2021.
  • M. M. Abed, U. Öztürk, and H. Khudhur, “Spectral CG Algorithm for Solving Fuzzy Non-linear Equations,” Iraqi J. Comput. Sci. Math., vol. 3, no. 1, 2022.
  • L. A. Zadeh, “Fuzzy sets,” Inf. Control, vol. 8, no. 3, pp. 338–353, 1965.
  • D. J. Dubois, Fuzzy sets and systems: theory and applications, vol. 144. Academic press, 1980.
  • H. Zimmermann, “Fuzzy set theory and its applications,” Int. Ser. Manag., 1991.
  • B. Hassan, H. Jabbar, and A. Al-Bayati, “A new class of nonlinear conjugate gradient method for solving unconstrained minimization problems,” 2019, doi: 10.1109/ICCISTA.2019.8830657.
  • B. A. Hassan, “A new type of quasi-newton updating formulas based on the new quasi-newton equation,” Numer. Algebr. Control Optim., vol. 10, no. 2, pp. 227–235, 2020, doi: 10.3934/naco.2019049.
  • WOLFE P, “CONVERGENCE CONDITIONS FOR ASCENT METHODS,” SIAM Rev., vol. 11, no. 2, pp. 226–235, 1969, doi: 10.1137/1011036.
  • H. M. Azzam, H. N. Jabbar, and K. K. Abo, “Four–Term Conjugate Gradient (CG) Method Based on Pure Conjugacy Condition for Unconstrained Optimization,” Kirkuk Univ. Journal-Scientific Stud., vol. 13, no. 2, pp. 101–113, 2018, doi: 10.32894/kujss.2018.145720.
  • Y. A. Laylani, K. K. Abbo, and H. M. Khudhur, “Training feed forward neural network with modified Fletcher-Reeves method,” J. Multidiscip. Model. Optim., vol. 1, no. 1, pp. 14–22, 2018, [Online]. Available: http://dergipark.gov.tr/jmmo/issue/38716/392124#article_cite.
  • A. S. Ahmed, “Optimization Methods For Learning Artificial Neural Networks,” University of Mosul, 2018.
  • A. S. Ahmed, H. M. Khudhur, and M. S. Najmuldeen, “A new parameter in three-term conjugate gradient algorithms for unconstrained optimization,” Indones. J. Electr. Eng. Comput. Sci., vol. 23, no. 1, 2021, doi: 10.11591/ijeecs.v23.i1.pp338-344.
  • H. M. Khudhur, “Numerical and analytical study of some descent algorithms to solve unconstrained Optimization problems,” University of Mosul, 2015.
  • K. K. Abbo and H. M. Khudhur, “New A hybrid conjugate gradient Fletcher-Reeves and Polak-Ribiere algorithm for unconstrained optimization,” Tikrit J. Pure Sci., vol. 21, no. 1, pp. 124–129, 2015.
  • K. K. Abbo and H. M. Khudhur, “New A hybrid Hestenes-Stiefel and Dai-Yuan conjugate gradient algorithms for unconstrained optimization,” Tikrit J. Pure Sci., vol. 21, no. 1, pp. 118–123, 2015.
  • K. K. ABBO, Y. A. Laylani, and H. M. Khudhur, “A NEW SPECTRAL CONJUGATE GRADIENT ALGORITHM FOR UNCONSTRAINED OPTIMIZATION,” Int. J. Math. Comput. Appl. Res., vol. 8, pp. 1–9, 2018.
  • K. K. Abbo, Y. A. Laylani, and H. M. Khudhur, “Proposed new Scaled conjugate gradient algorithm for Unconstrained Optimization,” Int. J. Enhanc. Res. Sci. Technol. Eng., vol. 5, no. 7, 2016.
  • Z. M. Abdullah, M. Hameed, M. K. Hisham, and M. A. Khaleel, “Modified new conjugate gradient method for Unconstrained Optimization,” Tikrit J. Pure Sci., vol. 24, no. 5, pp. 86–90, 2019.
  • H. M. Khudhur and K. K. Abbo, “A New Conjugate Gradient Method for Learning Fuzzy Neural Networks,” J. Multidiscip. Model. Optim., vol. 3, no. 2, pp. 57–69, 2020.
  • R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J., vol. 7, no. 2, pp. 149–154, 1964, doi: 10.1093/comjnl/7.2.149.
  • C. Witzgall and R. Fletcher, “Practical Methods of Optimization.,” Math. Comput., vol. 53, no. 188, p. 768, 1989, doi: 10.2307/2008742.
  • E. Polak and G. Ribiere, “Note sur la convergence de méthodes de directions conjuguées,” Rev. française d’informatique Rech. opérationnelle. Série rouge, vol. 3, no. 16, pp. 35–43, 1969, doi: 10.1051/m2an/196903r100351.
  • M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. (1934)., vol. 49, no. 6, p. 409, 1952, doi: 10.6028/jres.049.044.
  • Y. H. Dai and Y. Yuan, “A nonlinear conjugate gradient method with a strong global convergence property,” SIAM J. Optim., vol. 10, no. 1, pp. 177–182, 1999, doi: 10.1137/S1052623497318992.
  • Z.-J. Shi and J. Guo, “A new algorithm of nonlinear conjugate gradient method with strong convergence,” Comput. Appl. Math., vol. 27, pp. 93–106, 2008.
There are 37 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Mezher M. Abed1

Ufuk Öztürk 0000-0002-8800-7869

Hisham Mohammed 0000-0001-7572-9283

Publication Date December 15, 2021
Submission Date November 10, 2021
Published in Issue Year 2021 Volume: 2 Issue: 2

Cite

APA Abed1, M. M., Öztürk, U., & Mohammed, H. (2021). Spectral Fletcher (CD) Algorithm for Solving Fuzzy Non-Linear Equations. Journal of Soft Computing and Artificial Intelligence, 2(2), 108-119.
AMA Abed1 MM, Öztürk U, Mohammed H. Spectral Fletcher (CD) Algorithm for Solving Fuzzy Non-Linear Equations. JSCAI. December 2021;2(2):108-119.
Chicago Abed1, Mezher M., Ufuk Öztürk, and Hisham Mohammed. “Spectral Fletcher (CD) Algorithm for Solving Fuzzy Non-Linear Equations”. Journal of Soft Computing and Artificial Intelligence 2, no. 2 (December 2021): 108-19.
EndNote Abed1 MM, Öztürk U, Mohammed H (December 1, 2021) Spectral Fletcher (CD) Algorithm for Solving Fuzzy Non-Linear Equations. Journal of Soft Computing and Artificial Intelligence 2 2 108–119.
IEEE M. M. Abed1, U. Öztürk, and H. Mohammed, “Spectral Fletcher (CD) Algorithm for Solving Fuzzy Non-Linear Equations”, JSCAI, vol. 2, no. 2, pp. 108–119, 2021.
ISNAD Abed1, Mezher M. et al. “Spectral Fletcher (CD) Algorithm for Solving Fuzzy Non-Linear Equations”. Journal of Soft Computing and Artificial Intelligence 2/2 (December 2021), 108-119.
JAMA Abed1 MM, Öztürk U, Mohammed H. Spectral Fletcher (CD) Algorithm for Solving Fuzzy Non-Linear Equations. JSCAI. 2021;2:108–119.
MLA Abed1, Mezher M. et al. “Spectral Fletcher (CD) Algorithm for Solving Fuzzy Non-Linear Equations”. Journal of Soft Computing and Artificial Intelligence, vol. 2, no. 2, 2021, pp. 108-19.
Vancouver Abed1 MM, Öztürk U, Mohammed H. Spectral Fletcher (CD) Algorithm for Solving Fuzzy Non-Linear Equations. JSCAI. 2021;2(2):108-19.

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