Cyclic Gradient Methods for Unconstrained Optimization

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  • School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received date: 2021-09-29

  Revised date: 2022-06-22

  Online published: 2024-08-15

Supported by

This work is partly supported by the National Natural Science Foundation of China (Nos. 12171051 and 11871115).

Abstract

Gradient method is popular for solving large-scale problems. In this work, the cyclic gradient methods for quadratic function minimization are extended to general smooth unconstrained optimization problems. Combining with nonmonotonic line search, we prove its global convergence. Furthermore, the proposed algorithms have sublinear convergence rate for general convex functions, and R-linear convergence rate for strongly convex problems. Numerical experiments show that the proposed methods are effective compared to the state of the arts.

Cite this article

Ya Zhang, Cong Sun . Cyclic Gradient Methods for Unconstrained Optimization[J]. Journal of the Operations Research Society of China, 2024 , 12(3) : 809 -828 . DOI: 10.1007/s40305-022-00432-6

References

[1] Loosli, G., Canu, S.: chap. 5, pp. 111-135. John Wiley & Sons, Ltd (2010)
[2] Bottou, L., Curtis, F.E., Nocedal, J.: Optimization methods for large-scale machine learning. SIAM Rev. 60(2), 223-311(2018)
[3] Sun, C., Wang, Y.: Gravity-magnetic cross-gradient joint inversion by the cyclic gradient method. Optim. Method Softw. 35(5), 982-1001(2020)
[4] Ramillien, G., Frappart, F., Gratton, S., Vasseur, X.: Sequential estimation of surface water mass changes from daily satellite gravimetry data. J. Geod. 89(3), 259-282(2015)
[5] Larnier, S., Fehrenbach, J., Masmoudi, M.: The topological gradient method: From optimal design to image processing. Milan J. Math. 80(2), 411-441(2012)
[6] De Asmundis, R., Di Serafino, D., Hager, W.W., Toraldo, G., Zhang, H.: An efficient gradient method using the yuan steplength. Comput. Optim. Appl. 59(3), 541-563(2014)
[7] Huang, Y., Liu, H., Zhou, S.: Quadratic regularization projected Barzilai-Borwein method for nonnegative matrix factorization. Data Min. Knowl. Disc. 29(6), 1665-1684(2015)
[8] Jiang, B., Dai, Y.: Feasible Barzilai-Borwein-like methods for extreme symmetric eigenvalue problems. Optim. Method Softw. 28(4), 756-784(2013)
[9] Cauchy, A.: Méthode générale pour la résolution des systemes d’équations simultanées. Comput. Rend. Sci. Paris. 25, 536-538(1847)
[10] Akaike, H.: On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method. Ann. Inst. Stat. Math. Tokyo 11, 1-16(1959)
[11] Barzilai, J., Borwein, J.: Two-point step size gradient methods. IMA J. Numer. Anal. 8(1), 141-148(1988)
[12] Dai, Y., Yuan, Y.: Alternate minimization gradient method. IMA J. Numer. Anal. 23(3), 377-393(2003)
[13] Raydan, M.: On the BarzilaiandBorwein choice of steplength for the gradient method. IMA J. Numer. Anal. 13(3), 321-326(1993)
[14] Dai, Y., Liao, L.: R-linear convergence of the BarzilaiandBorwein gradient method. IMA J. Numer. Anal. 22(1), 1-10(2002)
[15] Dai, Y.: A new analysis on the Barzilai-Borwein gradient method. J. Oper. Res. Soc. China 1(2), 187-198(2013)
[16] Dai, Y.H., Hager, W.W., Schittkowski, K., Zhang, H.C.: The cyclic Barzilai-Borwein method for unconstrained optimization. IMA J. Numer. Anal. 26(3), 604-627(2006)
[17] Yuan, Y.: A new stepsize for the steepest descent method. J. Comput. Math. 24(2), 149-156(2006)
[18] Dai, Y., Yuan, Y.: Analysis of monotone gradient methods. J. Ind. Manag. Optim. 1(2), 181-192(2005)
[19] Dai, Y.: Alternate step gradient method. Optimization 52(4-5), 395-415(2003)
[20] Zhou, B., Gao, L.: Gradient methods with adaptive step-sizes. Comput. Optim. Appl. 35(1), 69-86(2006)
[21] Frassoldati, G., Zanni, L., Zanghirati, G.: New adaptive stepsize selections in gradient methods. J. Ind. Manag. Optim. 4(2), 299-312(2008)
[22] Fletcher, R.: A limited memory steepest descent method. Math. Program. 135, 413-336(2012)
[23] Gu, R., Du, Q.: A modified limited memory steepest descent method motivated by an inexact superlinear convergence rate analysis. IMA J. Numer. Anal. 41, 247-270(2020)
[24] De Asmundis, R., di Serafino, D., Hager, W., Zhang, H.: An efficient gradient method using the yuan steplength. Comput. Optim. Appl. 59(3), 541-563(2014)
[25] Sun, C., Liu, J.: New stepsizes for the gradient method. Optim. Lett. 14(4-5) (2020)
[26] Huang, Y., Dai, Y., Liu, X., Zhang, H.: Gradientmethods exploiting spectral properties. Optim.Method Softw. 35(4), 1-25(2020)
[27] Huang, Y., Liu, H.: Smoothing projected Barzilai-Borwein method for constrained non-lipschitz optimization. Comput. Optim. Appl. 65(3), 671-698(2016)
[28] Huang, Y., Dai, Y., Liu, X.: Equipping Barzilai-Borwein method with two dimensional quadratic termination property. SIAM J. Optim. 31(4), 3068-3096(2021)
[29] Liu, Z., Liu, H., Dong, X.: An efficient gradient method with approximate optimal stepsize for the strictly convex quadratic minimization problem. Optimization 67(3), 427-440(2018)
[30] Liu, Z., Liu, H.: An efficient gradient method with approximately optimal stepsize based on tensor model for unconstrained optimization. J. Optimiz. Theory. App. 181(2), 608-633(2019)
[31] Yuan, Y.: Step-sizes for the gradient method. Ams. Ip. Stud. Adv. Math. 42(2), 785-796(2008)
[32] De Asmundis, R., di Serafino, D., Riccio, F., Toraldo, G.: On spectral properties of steepest descent methods. IMA J. Numer. Anal. 33(4), 1416-1435(2013)
[33] Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for newton’s method. SIAM J. Numer. Anal. 23(4), 707-716(1986)
[34] Andrei, N.: An unconstrained optimization test functions collection. Adv. Model. Optim. 10(1), 147- 161(2008)
[35] Huang, Y., Liu, H.: On the rate of convergence of projected Barzilai-Borwein methods. Optim. Method Softw. 30(4), 880-892(2015)
[36] Raydan, M.: The BarzilaiandBorwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7(1), 26-33(1997)
[37] Toint, P.: Some numerical results using a sparse matrix updating formula in unconstrained optimization. Math. Comput. 32(143), 839-851(1978)
[38] Moré, J., Garbow, B., Hillstrom, K.: Testing unconstrained optimization software. ACM Trans. Math. Software 7, 17-41(1981)
[39] Bongartz,I., Conn,A.,Gould,N.,Toint, P.: CUTE: constrained and unconstrained testing environment. ACM Trans. Math. Softw. 21(1), 123-160(1995)
[40] Tibshirani, Ryan, J.: The lasso problem and uniqueness. Electron. J. Stat. 7(1), 1456-1490(2013)
[41] di Serafino, D., Ruggiero, V., Toraldo, G., Zanni, L.: On the steplength selection in gradient methods for unconstrained optimization. Appl. Math. Comput. 318, 176-195(2018)
[42] Dai, Y., Yuan, Y.: A nonlinear conjugate gradient method with a strong global convergence property. SIAM J. Optim. 10(1), 177-182(1999)
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