Application of a globally convergent hybrid conjugate gradient method in portfolio optimization. (English) Zbl 07877106
Summary: In this paper, we propose a modification that improves efficiency, robustness and reliability of the famous HS conjugate gradient method. In particular, we propose a hybrid of the HS and DHS methods, where DHS is another recent modification of the HS method. Irrespective of the line search, the search direction of the proposed method is sufficiently descent. Moreover, the new approach guarantees global convergence for general functions under the strong Wolfe line search. Numerical results and performance profiles are reported, and indicate that the new approach outperforms three similar methods in the literature. We also give a practical application of the new approach in minimizing risk in portfolio selection.
MSC:
| 90C06 | Large-scale problems in mathematical programming |
| 90C30 | Nonlinear programming |
| 65K05 | Numerical mathematical programming methods |
Keywords:
conjugate gradient; global convergence; strong Wolfe-Powell conditions; portfolio selectionReferences:
| [1] | A. B. Abubakar, P. Kumam, M. Malik, P. Chaipunya and A. H. Ibrahim, A hybrid FR-DY conjugate gradient algorithm for unconstrained optimization with application in portfolio selection. AIMS Math, 6(6) (2021), 6506-6527. · Zbl 1484.90138 |
| [2] | O. J. Adeleke, M. O. Olusanya and I. A. Osinuga, A PRP-HS Type Hybrid Nonlinear Conjugate Gradient Method for Solving Unconstrained Optimization Problems, In: Silhavy, R., Silhavy, P., Prokopova, Z (eds) Intelligent Systems Applications in Software Engineering. CoMeSySo 2019 (2019). Advances in Intelligent Systems and Computing, 1046, Springer, Cham. |
| [3] | A. Alhawarat, M. Mamat, M. Rivaie and Z. Salleh, An efficient hybrid conjugate gradient method with the strong Wolfe-Powell line search, Math. Probl. Eng., 2015: 103517 (2015). · Zbl 1394.90556 |
| [4] | N. Andrei, An unconstrained optimization test functions collection, Adv. Model. Optim., 10(1) (2008), 147-161. · Zbl 1161.90486 |
| [5] | A. M. Awwal, I. M. Sulaiman, M. Malik, M. Mamat, P. Kumam and K. Sitthithakerngkiet, A spectral RMIL+ conjugate gradient method for unconstrained optimization with applications in portfolio selection and motion control. IEEE Access, 9 (2021), 75398-75414. |
| [6] | M. C. Bartholomew-Biggs, Nonlinear Optimization with Financial Applications. Kluwer Academic Publishers, Boston (2006). |
| [7] | M. C. Bartholomew-Biggs and S. J. Kane, A global optimization problem in portfolio selection. Comput. Manag. Sci., 6 (2009) 329-345. · Zbl 1187.91193 |
| [8] | Y. H. Dai, A family of hybrid conjugate gradient methods for unconstrained optimization, Math. Comput., 72(243) (2003), 1317-1328. · Zbl 1014.49024 |
| [9] | Y. H. Dai and L. Liao, New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim., 43 (2001), 87-101. · Zbl 0973.65050 |
| [10] | Z. Dai and F. Wen, Another improved Wei-Yao-Liu nonlinear conjugate gradient method with sufficient descent property, Appl. Math. Comput., 218(14) (2012), 7421-7430. · Zbl 1254.65074 |
| [11] | T. Diphofu and P. Kaelo, Another three-term conjugate gradient method close to the memoryless BFGS for large scale unconstrained optimization problems, Mediterr. J Math., 18:211 (2021). · Zbl 1477.90040 |
| [12] | T. Diphofu, P. Kaelo and A. R. Tufa, A modified nonlinear conjugate gradient algorithm for unconstrained optimization and portfolio selection problems. RAIRO Oper. Res., 57(2) (2023), 817-835. · Zbl 1519.90113 |
| [13] | E. D. Dolan, J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-214. · Zbl 1049.90004 |
| [14] | M. Fang, M. Wang, M. Sun and R. Chen, A Modified Hybrid Conjugate Gradient Method for Unconstrained Optimization, J. Math., 2021(1) (2021), 1-9. · Zbl 1477.90102 |
| [15] | R. Fletcher, Practical methods of optimization, Unconstrained Optimization, John Wiley & Sons, New York 1(1987). · Zbl 0905.65002 |
| [16] | R. Fletcher and C. M. Reeves, Function minimization by conjugate gradients, Computer J., 7(2) (1964), 149-154. · Zbl 0132.11701 |
| [17] | H. Guan and S. Wang, A Modified Conjugate Gradient Method for Solving Large-Scale Nonlinear Equations, Math. Probl. Eng., 2021:9919595 (2021), doi:10.1155/2021/9919595. · Zbl 1512.90219 |
| [18] | W. W. Hager and H. Zhang, A survey of nonlinear conjugate gradient methods, Pacific J. Optim., 2 (2011), 35-58. · Zbl 1117.90048 |
| [19] | M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J Res. Nat. Bureau Stand., 49(6) (1952), 409-436. · Zbl 0048.09901 |
| [20] | X. Jiang, J. Jian, D. Song and P. Liu, An improved Polak-Ribiére-Polyak conjugate gradient method with an efficient restart direction, Comput. Appl. Math., 40(174) (2021), 1-24. · Zbl 1476.90258 |
| [21] | X. Jiang, W. Liao, J. Yin, J. Yin and J. Jian, A new family of hybrid three-term conjugate gradient methods with applications in image restoration, Numer. Algor., 91(2) (2022), 1-31. · Zbl 1496.65076 |
| [22] | Y. Liu and C. Storey, Efficient generalized conjugate gradient algorithms, Part 1: Theory, J Optim. Theory Appl., 69(1) (1991), 129-137. · Zbl 0702.90077 |
| [23] | H. Markowitz, Portfolio Selection, J. Finance, 7(1) (1952), 77-91. |
| [24] | P. Mtagulwa and P. Kaelo, An efficient modified PRP-FR hybrid conjugate gradient method for solving unconstrained optimization problems, Appl. Numer. Math., 145 (2019), 111-120. · Zbl 1477.65095 |
| [25] | A. A. Mustafa, New Spectral LS conjugate gradient method for nonlinear unconstrained optimization, Int. J Comput. Math.,100 (2023), 838-846. · Zbl 1513.65189 |
| [26] | E. Polak and G. Ribiére, Note sur la convergence de directions conjugées, Rev. Francaise Informat Recherche Opèrationelle, 3e Année, 16 (1969), 35-43. · Zbl 0174.48001 |
| [27] | B. T. Polyak, The conjugate gradient method in extreme problems, USSR Comput. Math. Math. Phys, 9(4) (1969), 94-112. · Zbl 0229.49023 |
| [28] | I. M. Sulaiman, M. Malik, A. M. Awwal, P. Kumam, M. Mamat and S. Al-Ahmad, On three-term conjugate gradient method for optimization problems with applications on COVID-19 model and robotic motion control, Adv Cont Discr Mod., 2022:1 (2022). doi:10.1186/s13662-021-03638-9. |
| [29] | Z. Wei, S. Yao and L. Liu, The convergence properties of some new conjugate gradient methods, Appl. Math. Comput., 183(2) (2006), 1341-1350. · Zbl 1116.65073 |
| [30] | L. Zhang, An improved Wei-Yao-Liu nonlinear conjugate gradient method for optimization computation, Appl. Math. Comput., 215(6) (2009), 2269-2274. · Zbl 1181.65089 |
| [31] | G. Zoutendijk, Nonlinear programming, computational methods, In: J. Abadie Ed., Integer and Nonlinear Programming, North-Holland, Amsterdam, 1970, pp. 37-86. · Zbl 0336.90057 |
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