An Efficient Hybrid Conjugate Gradient Algorithm for Solving Intuitionistic Fuzzy Nonlinear Equations

(Umar Audu Omesa , Ibrahim Mohammed Sulaiman , Waziri Muhammad Yusuf , Basim A. Hassan, Aliyu Usman Moyi, Ayu Abdul-Rahman and Mustafa Mamat)

---

Abstract

This paper presents an iterative algorithm for solving intuitionistic fuzzy nonlinear equations (IFNEs). The proposed method is based on the classical conjugate gradient (CG) search direction. An interesting feature of the new algorithm is that it considers problems based on the special triangular intuitionistic fuzzy number. For this purpose, intuitionistic fuzzy quantities are transformed into membership and non-membership parametric forms, and a line search procedure is employed to compute the step length. Preliminary results from numerical experiments are presented to demonstrate the performance of the method. It is observed that the proposed hybrid CG method is highly effective and promising.
KEYWORDS
Hybrid CG, intuitionistic fuzzy nonlinear equation, parametric form, step length, inexact line search
PDF

References

Abbasbandy, S. and Asady, B. (2004). Newton's method for solving fuzzy nonlinear equations. Applied Mathematics and Computation, 159(2), 349–56.
Al-Baali, M. (1985). Descent property and global convergence of the Fletcher-Reeves method with inexact line search. IMA Journal of Numerical Analysis, 5(1), 121–4.
Amma, B.B., Melliani, S. and Chadli, L.S. (2016). Numerical solution of intuitionistic fuzzy differential equations by Euler and Taylor methods. Notes on Intuitionistic Fuzzy Sets, 22(2), 71–86.
Andrei, N. (2008). Another hybrid conjugate gradient algorithm for unconstrained   optimization. Numerical Algorithms, 47(2), 143–56.
Atanassov, K.T. (1999). Intuitionistic fuzzy sets. In: K.T. Atanassov (Ed.). Intuitionistic Fuzzy Sets Studies in Fuzziness and Soft Computing. Physica, Heidelberg.
Audu, O.U., Mamat, M., Sulaiman, I.M., Waziri, M.Y. and Abba, V.M. (2020). A new modification of conjugate gradient parameter with efficient line search for nonconvex function. International Journal of Scientific and Technology Research, 9(3),1–4.
Biswas, S., Banerjee, S. and Roy, T.K. (2016). Solving intuitionistic fuzzy differential equations with linear differential operator by Adomian decomposition method. Notes on Intuitionistic Fuzzy Sets, 22(4), 25–41.
Buckley, J.J. and Qu, Y. (1990). Solving linear and quadratic fuzzy equations. Fuzzy sets and systems, 38(1), 43–59.
Buckley, J.J. and Qu, Y. (1991). Solving fuzzy equations: A new solution concept. Fuzzy Sets and Systems, 39(3), 291–301
Dai, Y.H. and Yuan Y.Y. (1999). A nonlinear conjugate gradient with strong global convergence properties. SIAM Journal on Optimization, 10(1), 177–82.
Dai, Y.H. and Yuan, Y. (2001). An efficient hybrid conjugate gradient method for unconstrained optimization. Annals of Operations Research, 103(1–4), 33–47.
Dubois, D.J. (1980). Fuzzy Sets and Systems: Theory and Applications. Amsterdam: Elsevier, Academic press.
Ettoussi, R., Melliani, S., Elomari, M. and Chadli, L.S. (2015). Solution of intuitionistic fuzzy differential equations by successive approximations method. Notes on Intuitionistic Fuzzy Sets, 21(2), 51–62.
Fletcher, R. and Reeves, C. (1964). Function minimization by conjugate gradients. The Computer Journal, 7(2), 149–54. Doi: 10.1093/comjnl/7.2.149.
Gilbert, J.C. and Nocedal, J. (1992). Global convergence properties of conjugate gradient methods for optimization. SIAM Journal on Optimization, 2(1), 21–42.
Goetschel, Jr. R. and Voxman, W. (1986). Elementary fuzzy calculus. Fuzzy sets and systems, 18(1), 31–43.
Hestenes, M.R. and Stiefel, E.L. (1952). Methods of conjugate gradients for solving linear systems. J. Research Nat.Bur. Standards, 49(6), 409–36. Doi:10.6028/JRES.049.044.
Kajani, M.T. Asady, B. and Vencheh, A.H. (2005). An iterative method for solving dual fuzzy nonlinear equations. Applied Mathematics and Computation, 167(1), 316–23.
Keyanpour, M. and Akbarian, T. (2014). Solving intuitionistic fuzzy nonlinear equations. Journal of Fuzzy Set Valued Analysis, 2014(n/a), 1–6.
Liu, J.K. and Li, S.J. (2014). New hybrid conjugate gradient method for unconstrained optimization. Applied Mathematics and Computation, 245(n/a), 36–43.
Malik, M., Mamat, M., Abas, S.S., Ibrahim, S.M. and Sukono. (2020).  A new spectral conjugate gradient method with descent condition and global convergence property for unconstrained optimization. Journal of Mathematical and Computational Science, 10(5), 2053–69.
Malik, M., Mamat, M., Abas, S.S., Ibrahim, S.M. and Sukono. (2021). Performance analysis of new spectral and hybrid conjugate gradient methods for solving unconstrained optimization problems, IAENG. Int. J. Comput. Sci., 48, 66–79. 
Mohammed, S.I., Mamat, M., Ghazali, P.L., Foziah, H.M, Kamfa, K., Pinontoan, B. and Rindengan, A.J. (2020).  An improved shamanskii method for solving nonlinear equation. Journal of Advanced Research in Dynamical and Control Systems, 12(2), 591–4.
Polak, E. and Ribiere, G. (1969). Note on the convergence of conjugate directions (in French). Revue Française D'informatique Et De Recherche Opérationnelle, 3(R1), 35–43.
Polyak, B.T. (1969). The conjugate gradient method in extremal problems. USSR Computational Mathematics and Mathematical Physics, 9(4), 94–112. Doi: 10.1016/0041-5553(69)90035-4.
Shaw, A.K. and Roy, T.K. (2012). Some arithmetic operations on triangular intuitionistic fuzzy number and its application on reliability evaluation. International Journal of Fuzzy Mathematics and Systems, 2(4), 363–82.
Sulaiman, I.M. and Mamat, M. (2020a). A new conjugate gradient method with descent properties and its application to regression analysis. J. Numer. Anal. Ind. Appl. Math., 14(n/a), 25–39. 
Sulaiman, I.M., Malik, M., Awwal, A.M., Kumam, P., Mamat, M. and Al-Ahmad, S. (2022a). On three-term conjugate gradient method for optimization problems with applications on COVID-19 model and robotic motion control. Adv Cont Discr Mod. 1(n/a). https://doi.org/10.1186/s13662-021-03638-9
Sulaiman, I.M., Mamat, M. and Ghazali, P.L. (2021). Shamanskii method for solving parameterized fuzzy nonlinear equations. International Journal of Optimization and Control: Theories and Applications, 11(1), 24–9. 
Sulaiman, I.M., Mamat, M., Malik, M., Kottakkaran, S.N. and Ashraf, E. (2022b). Performance analysis of a modified Newton method for parameterized dual fuzzy nonlinear equations and its application. Results in Physics, 33(n/a), 105140,
Sulaiman, I.M., Mamat, M., Waziri, M.Y., Fadhilah, A. and Kamfa, K.U. (2016).  Regula falsi method for solving fuzzy nonlinear equation. Far East Journal of Mathematical Sciences, 100(6), 873–84
Sulaiman, I.M., Waziri, M.Y., Olowo, E.S. and Talat, A.N. (2018). Solving fuzzy nonlinear equations with a new class of conjugate gradient method. Malaysian Journal of Computing and Applied Mathematics, 1(1), 11-9. 
Touati-Ahmed, D. and Storey, C. (1990). Efficient hybrid conjugate gradient techniques. Journal of Optimization Theory and Applications, 64(2), 379–97.
Umar, A.O., Sulaiman, I.M., Mamat, M., Waziri, M.Y., Foziah, H.M., Altien, J. R. and Deiby, T.S. (2020a). New hybrid conjugate gradient method for solving fuzzy nonlinear equations. Journal of Advanced Research in Dynamical and Control Systems, 12(2), 585–90.
Umar, A.O., Waziri, M.Y. and Sulaiman, I.M. (2018). Solving dual fuzzy nonlinear equations via a modification of shamanskii steps. Malaysian Journal of Computing and Applied Mathematics, 1(2), 1–9.
Waziri, M.Y. and Moyi, A.U. (2016). An alternative approach for solving dual fuzzy nonlinear equations. International Journal of Fuzzy Systems, 18(1), 103-7.
Yakubu, U.A., Sulaiman, I.M., Mamat, M., Ghazali, P. and Khalid, K. (2020). The global convergence properties of a descent conjugate gradient method. Journal of Advanced Research in Dynamical and Control Systems, 12(2), 1011–6.
Yang, L., Ji-Xue, H. and Hong-yan, Y. (2008). Normal technique for ascertaining nonmembership functions of Intuitionistic Fuzzy Sets. In: 2008 Chinese Control and Decision Conference, Yantai, Shandong, 02-04/07/2008.
Zadeh, L.A. (1996). Fuzzy sets, fuzzy logic, and fuzzy systems: selected papers by Lotfi A Zadeh. In: G.J. Klir, B. Yuan (eds.). Advances in Fuzzy Systems - Applications and Theory. USA: State University of New York.