Fuzzy ϑ –BFGS Update for Numerical Optimization

Saad Shaker Mahmood, Alan Jalal Abdulqader

Abstract


In this paper, we develop a quasi-Newton method for unconstrained optimization problems with fuzzy functions. Also, we Propose  the -BFGS method to fuzzy optimization problems. The generalized Hukuhara differentiability for fuzzy functions is employed. By using a Hessian approximation, we resolve the high computational cost of finding the Hessian in Newton method for fuzzy optimization problems. We will find the solutions of a fuzzy optimization problem. Finally, some numerical results support this claim and also indicate that the -BFGS update may be competitive with the BFGS update in general

Keywords


Unconstrained fuzzy optimization problems, Quasi-Newton methods, Superlinear convergence, generalized Hukuhara differentiability,

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References


Arana-Jim´enez, M., Rufi´an-Lizana, A., Chalco-Cano, Y., and H. Rom´an-Flores, Generalized convexity in fuzzy vector optimization through a linear ordering, Information Sciences, vol.312, pp.13–24, 2015.

Bazaraa, M.S., Sherali, H.D., and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, 2013.

Bector, C.R., and S. Chandra, Fuzzy Mathematical Programming and Fuzzy Matrix Games, Springer-Verlag, Berlin, 2005.

Bede, B., and L. Stefani, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, vol.230, pp.119–141, 2013.

Bellman, R.E., and L.A. Zadeh, Decision making in a fuzzy environment, Management Science, vol.17, pp.141–164, 1970.

Chalco-Cano, Y., Silva, G.N., and A. Rufian-Lizana, On the Newton method for solving fuzzy optimization problems, Fuzzy Sets and Systems, vol.272, pp.60–69, 2015.

Fliege, J., Grana Drummond, L.M., and B.F. Svaiter, Newton’s method for multiobjective optimization, SIAM on Optimization, vol.20, no.2, pp.602–626, 2009.

Ganesan, K., and P. Veeramani, Fuzzy linear programming with trapezoidal fuzzy numbers, Annals of Operations Research, vol.143, pp.305–315, 2006.

Ghaznavi, M., Soleimani, F., and N. Hoseinpoor, Parametric analysis in fuzzy number linear programming prob- lems, International of Fuzzy Systems, vol.18, no.3, pp.463–477, 2016.

Ghosh, D., A quasi-Newton method with rank-two update to solve interval optimization problems, International of Applied and Computational Mathematics, 2016, doi:10.1007/s40819-016-0202-7.

Ghosh, D., A Newton method for capturing efficient solutions of interval optimization problems, OPSEARCH, vol.53, no.3, pp.648–665, 2016.

Ghosh, D., Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions, of Applied Mathematics and Computing, 2016, doi:10.1007/s12190-016-0990-2.

Griva, I., Nash, S.G., and A. Sofer, Linear and Nonlinear Optimization, 2nd Edition, SIAM, 2009.

Hosseinzadeh Lotfi, F., Allahviranloo, T., Alimardani Jondabeh, M., and L. Alizadeh, Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution, Applied Mathematical Modelling, vol.33, pp.3151–3156, 2009.

Lodwick, W.A., and J. Kacprzyk (Eds.), Fuzzy Optimization: Recent Advances and Applications, Springer-Verlag, Berlin, 2010.

Mahdavi-Amiri, N., and S.H. Nasseri, Duality in fuzzy number linear programming by use of a certain linear ranking function, Applied Mathematics and Computation, vol.180, pp.206–216, 2006.

Maleki, H.R., Ranking functions and their applications to fuzzy linear programming, Far East Mathe- matics Sciences, vol.4, pp.283–301, 2002.

Mottaghi, A., Ezzati, R., and E. Khorram, A new method for solving fuzzy linear programming problems based on the fuzzy linear complementary problem (FLCP), International of Fuzzy Systems, vol.17, no.2, pp.236– 245, 2015.

Nocedal, J., and S. Wright, Numerical Optimization, Springer Science and Business Media, 2006.

Pirzada, U.M., and V.D. Pathak, Newton method for solving the multi-variable fuzzy optimization problem,

Qu, S., Goh, M., and F.T.S. Chan, Quasi-Newton methods for solving multiobjective optimization, Operations Research Letters, vol.39, pp.397–399, 2011.

Stefanini, L., A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and Systems, vol.161, no.11, pp.1564–1584, 2010.

Stefanini, L., and B. Barnabas, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis: Theory, Methods & Applications, vol.71, no.3, pp.1311–1328, 2009.

Saad Shakir M, Ali Ibrahim M and Balasim Thaha A α-BFGS update for unconstrained optimization, of College of Education No.1.2011

Tanaka, H., Okuda, T., and K. Asai, On fuzzy mathematical programming, of Cybernetics, vol.3, pp.37– 46, 1974.




DOI: http://dx.doi.org/10.52155/ijpsat.v6.1.175

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