A New Non-monotone Line Search Algorithm to Solve Non-smooth Optimization Finance Problem
الموضوعات :Saeed Banimehri 1 , Hamid Esmaeili 2
1 - Department of Mathematics, Bu-Ali sina University, hamedan, Iran
2 - Department of Mathematics, Bu-Ali Sina University, Hamedan, Iran
الکلمات المفتاحية: Derivative-free optimization, Non-smooth optimization, Non-monotone Armijo line search, Diagonal discrete gradient bundle method,
ملخص المقالة :
In this paper, a new non-monotone line search is used in the diagonal discrete gradient bundle method to solve large-scale non-smooth optimization problems. Non-smooth optimization problems are encountered in many applications in fi-nance problems. The new principle causes the step in each iteration to be longer, which reduces the number of iterations, evaluations, and the computational time. In other words, the efficiency and performance of the method are improved. We prove that the diagonal discrete gradient bundle method converges with the pro-posed non-monotone line search principle for semi-smooth functions, which are not necessarily differentiable or convex. In addition, the numerical results confirm the efficiency of the proposed correction.
[1] Ahookhosh, M., Amini, K., Bahrami, S., A class of nonmonotone Armijo-type line search method for unconstrained optimization, Journal of Optimization Theory and Applications, 2012; 61(4): 387-404. doi: 10.1080/02331934.2011.641126
[2] Bagirov, A.M., Karasözen, B., Sezer, M., Discrete gradient method: derivative-free method for nonsmooth optimization, Journal of Optimization Theory and Applications, 2008; 137(2):317-334. doi: 10.1007/s10957-007-9335-5
[3] Bagirov, A.M., Karmitsa, N., Mäkelä, M.M., Introduction to Nonsmooth Optimization: theory, practice and software, Vol. 12. Springer, Switzerland, 2014.
[4] Barzilai, J., Borwein, J.M., Two-point step size gradient methods, IMA journal of numerical analysis, 1988; 8(1): 141-148. doi: 10.1093/imanum/8.1.141
[5] Dai, Y.H., Liao, L.Z., R‐linear convergence of the Barzilai and Borwein gradient method, IMA Journal of Numerical Analysis, 2002; 22(1):1-10. doi: 10.1093/imanum/22.1.1
[6] Dai, Y.H., On the nonmonotone line search, Journal of Optimization theory and Applications, 2002; 112(2): 315-330. doi: 10.1023/A:1013653923062
[7] Darabi, R., Baghban, M., Application of Clayton Copula in Portfolio Optimization and its Comparison with Markowitz Mean-Variance Analysis, Advances in mathematical finance & applications, 2018; 3(1):33-51. doi: 10.22034/amfa.2018.539133
[8] Grimaldi, R., Fibonacci and Catalan Numbers, John Wiley and Sons, 2012.
[9] Grippo, L., Lampariello, F., Lucidi, S., A class of nonmonotone stabilization methods in unconstrained optimization, Numerische Mathematik, 1991; 59(1): 779-801. doi: 10.1007/BF01385810
[10] Grippo, L., Lampariello, F., Lucidi, S., A nonmonotone line search technique for Newton’s method, SIAM Journal of Numerical Analysis, 1986; 23(4): 707-716. doi: 10.1137/0723046
[11] Griva, I., Nash, S.G., Sofar, A., Linear and nonlinear optimization, Vol. 108. SIAM, Philadelphia, 2009.
[12] Haarala, M., The Large-scale nonsmooth optimization: variable metric bundle method with limited memory, Academic Jyvaskyla, Jyvaskyla, 2004.
[13] Haarala, M., Miettinen, K., Makela, M.M., New limited memory bundle method for large-scale nonsmooth optimization, Optimization Methods and Software, 2004;19(6):673-692. doi: 10.1080/10556780410001689225
[14] Haarala, N., Miettinen, K., Makela, M.M., Globally convergent limited memory bundle method for large-scale nonsmooth optimization, Mathematical Programming, 2007; 109(1):181-205. doi: 10.1007/s10107-006-0728-2
[15] Karmitsa, N., Diagonal bundle method for nonsmooth sparse optimization, Journal of Optimization Theory and Applications, 2015; 166(3):886-905. doi: 10.1007/s10957-014-0666-8
[16] Karmitsa, N., Diagonal discrete gradient bundle method for derivative free nonsmooth optimization, Optimization, 2016; 65(8):1599-1614. doi: 10.1080/02331934.2016.1171865
[17] Lemarechal, C., Strodiot, J.J., Bihain, A., On a bundle algorithm for nonsmooth optimization, Nonlinear programming4; 1981: 245-282. doi: 10.1016/B978-0-12-468662-5.50015-X
[18] Mifflin, R., A modification and an extension of Lemaréchal’s algorithm for nonsmooth minimization, Nondifferential and variational techniques in optimization,1980; 77-90.doi: 10.1007/BFb0120960
[19] Rahmani, M., Khalili Eraqi, M., Nikoomaram, H., Portfolio Optimization by Means of Meta Heuristic Algorithms, Advances in mathematical finance & applications, 2019; 4(4):83-97. doi:10.22034/amfa.2019.579510.1144
[20] Raydan, M., On the Barzilai and Borwein choice of steplength for the gradient method, IMA Journal of Numerical Analysis, 1993; 13(3): 321-326. doi:10.1093/imanum/13.3.321
[21] Raydan, M., The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM Journal on Optimization, 1997; 7(1):26-33. doi:10.1137/S1052623494266365
[22] Sun, W., Han, J., Sun, J., Global convergence of nonmonotone descent methods for unconstrained optimization problems, Journal of Computational and Applied Mathematics, 2002; 146(1):89-98. doi: 10.1016/S0377-0427(02)00420-X
[23] Vlcek, J., Lukšan, L., Globally convergent variable metric method for nonconvex nondifferentiable unconstrained minimization, Journal of Optimization Theory and Applications, 2001; 111(2):407-430. doi: 10.1023/A:1011990503369
[24] Zanjirdar, M., Overview of Portfolio Optimization Models, Advances in mathematical finance & applications, 2020; 5(4):419-435. doi:10.22034/amfa.2020.1897346.1407
[25] Zhang, H., Hager, W.W., A nonmonotone line search technique and its application to unconstrained optimization, SIAM journal on Optimization, 2004;14(4):1043-1056. doi: 10.1137/S1052623403428208