Barzilai-Borwein gradient method for solving fuzzy nonlinear equations
Keywords:Barzilai and Borwein method, gradient, Fuzzy nonlinear equations, parametric form,
In this paper, we employ a two-step gradient method for solving fuzzy nonlinear equations. This method is Jacobian free and only requires a line search for 15k=0"> . The fuzzy coefficients are presented in parametric form. Numerical experiments on well-known benchmark problems have been presented to illustrate the efficiency of the proposed method.
 H. J. Zimmermann, Fuzzy Set Theory and its Applications, third ed., Kluwer Academic, Norwell, MA, 1991.
 L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
 M. Waziri, A. Moyi, An alternative approach for solving dual fuzzy nonlinear equations. International Journal of Fuzzy Systems, 18 (2016), 103 â€“ 107.
 I.M. Sulaiman, M. Mamat, M.A. Mohamed, M.Y. Waziri, Diagonal Updating Shamasnkii-Like method for Solving Fuzzy Nonlinear Equation. Far East Journal of Math Sci 103, 10, (2018) 1619-1629.
 J. Barzilai, J. M. Borwein, Two-Point step size gradient methods. IMA Journal of Numerical Analysis, 8(1988), 141-148.
 J.J. Buckley, Y. Qu, Solving fuzzy equations: A new solution concept, Fuzzy Set and Systems, 39 (1991), 291- 301.
 J.J. Buckley, Y. Qu, Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems 38 (1990) 43 - 59.
 S. Wenyu and Y. Ya-Xiang, Optimization theory and methods, Springer Optimization and Its Applications (2006).
 S, Abbasbandy, A. Jafarian, Steepest descent Method for solving fuzzy nonlinear equations. Applied Mathematics and Computation, 174 (2006), 669 â€“ 675.
 S, Abbasbandy, B. Asady, Newton Method for solving fuzzy nonlinear equations. Applied Mathematics and Computation, 159 (2004), 349 â€“ 356.
 I. M. Sulaiman, M. Mamat, M. Y. Waziri, A. Fadhilah, K. U. Kamfa: Regula Falsi Method for Solving Fuzzy Nonlinear Equation. Far East Journal of Math Sci., 100, 6, (2016), 873-884.
 R. Amirah, M., Lazim, M. Mamat, Broydenâ€™s method for solving Fuzzy nonlinear equations. Advances in Fuzzy System, 2010, (2010) Article ID 763270, 6 pages.
 D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Application, Academic Press, New York, 1980.
 S. Muzzioli, H. Reynaerts, Fuzzy linear systems of the form A1x + b1 = A2x + b2, Fuzzy Sets and Systems, 157, 7, (2006), 939â€“951.
 R. Goetschel Jr, W. Voxman, â€œElementary fuzzy calculus,â€ Fuzzy set and Systems, 18, (1986), 31-43.
 E. K. P. Chong and S. H. Zak, An Introduction to Optimization, Wiley Series in Discrete Mathematics and Optimization (2013).
 Y.H. Dai, Y.J. Yuan, and Y. Yuan, Modified two-point step size gradient methods for unconstrained optimization, Comput. Optim. Appl., 22, 1, (2002), 103 109.
 M. Raydan. The Barzilai and Borwien gradient method for the largescale unconstrained minimization problem, SIAM J. Optim., 7, 1, (1997), 26 33.
 S. Babaie-Kafaki and M. Fatemi, A modified two-point step size gradient algorithm for unconstrained minimization, Optimization Methods and Software, 28, 5, (2013) 1040-1050.
 M. Raydan, On the Barzilai and Borwein choice of step length for the gradient method, IMA J. Numer. Anal., 13 (1993), 321â€“326.
 I.M. Sulaiman, M. Mamat, M.Y. Waziri, M.A. Mohamed, F.S. Mohamad. Solving Fuzzy Nonlinear Equation via Levenberg-Marquardt Method. Far East Journal of Math Sci, 103, 10, (2018), 1547-1558.
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