Abstract
In the framework of large-scale optimization problems, the standard BFGS method is not affordable due to memory constraints. The so-called limited-memory BFGS (L-BFGS) method is an adaption of the BFGS method for large-scale settings. However, the standard BFGS method and therefore the standard L-BFGS method only use the gradient information of the objective function and neglect function values. In this paper, we propose a new regularized L-BFGS method for solving large scale unconstrained optimization problems in which more available information from the function and gradient values are employed to approximate the curvature of the objective function. The proposed method utilizes a class of modified quasi-Newton equations in order to achieve higher order accuracy in approximating the second order curvature of the objective function. Under some standard assumptions, we provide the global convergence property of the new method. In order to provide an efficient method for finding global minima of a continuously differentiable function, a hybrid algorithm that combines a genetic algorithm (GA) with the new proposed regularized L-BFGS method has been proposed. This combination leads the iterates to a stationary point of the objective function with higher chance of being global minima. Numerical results show the efficiency and robustness of the new proposed regularized L-BFGS and its hybridized version with GA in practice.
Similar content being viewed by others
References
Al-Baali, M.: Improved Hessian approximations for limited memory BFGS method. Numer. Algorithms 22, 99–112 (1999)
Al-Baali, M., Grandinetti, L., Pisacane, O.: Damped techniques for the limited memory BFGS method for large-scale optimization. J. Optim. Theory Appl. 161(2), 688–699 (2014)
Andrei, N.: An unconstrained optimization test functions collection. Adv. Model. Optim. 10(1), 147–161 (2008)
Babaie-Kafaki, S., Ghanbari, R., Mahdavi-Amiri, N.: Two new conjugate gradient methods based on modified secant equations. J. Comput. Appl. Math. 234, 1374–1386 (2010)
Biglari, F., Hassan, M.A., Leong, W.J.: New quasi Newton methods via higher order tensor models. J. Comput. Appl. Math. 235, 2412–2422 (2011)
Broyden, C.G., Dennis, J.E., Moré, J.J.: On the local and superlinear convergence of quasi-Newton methods. IMA J. Appl. Math. 12(3), 223–245 (1973)
Byrd, R.H., Lu, P.H., Nocedal, J., Zhu, C.Y.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16, 1190–1208 (1995)
Byrd, R.H., Nocedal, J.: A tool for the analysis of quasi-Newton methods with application to unconstrained minimization. SIAM J. Numer. Anal. 26(3), 727–739 (1989)
Byrd, R.H., Nocedal, J., Schnabel, R.B.: Representations of quasi-Newton matrices and their use in limited memory methods. Math. Program. A 63(2), 129–156 (1994)
Byrd, R.H., Nocedal, J., Yuan, Y.: Global convergence of a class of quasi Newton methods on convex problems. SIAM J. Numer. Anal. 24(5), 1171–1190 (1987)
Chen, X.J.: Convergence of the BFGS method for LC1 convex constrained optimization. SIAM J. Control Optim. 34, 2051–2063 (1996)
Dai, Y.: Convergence properties of the BFGS algorithm. SIAM J. Optim. 13(3), 693–701 (2002)
Davis, L.: The Handbook of Genetic Algorithms. Van Nostrand Reingold, New York (1991)
Dembo, R., Steihaug, T.: Truncated Newton algorithms for large-scale unconstrained optimization. Math. Program. 26(2), 190–212 (1983)
Dennis, J.E., Moré, J.J.: Quasi-Newton methods, motivation and theory. SIAM Rev. 19, 46–89 (1977)
Dolan, E.D., Moré, J.J.: Benchmarking optimizations of tware with performance profiles. Math. Program. 91(2), 201–203 (2002)
Fletcher, R.: Practical Methods of Optimization. A Wiley-Interscience Publication, 2nd edn. Wiley, Chichester (1987)
Gould, N., Orban, D., Toint, P.L.: CUTEr, a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29(4), 373–394 (2003)
La Cruz, W., Noguera, G.: Hybrid spectral gradient method for the unconstrained minimization problem. J. Glob. Optim. 44(2), 193–212 (2009)
Li, D., Fukushima, M.: A modified BFGS method and its global convergence in non convex minimization. J. Comput. Appl. Math. 129, 15–35 (2001)
Li, B., Ong, Y.S., Le, M.N., Goh, C.K.: Memetic gradient search. In: Evolutionary Computation, IEEE World Congress on Computational Intelligence, pp. 2894–2901 (2008)
Liu, T.W.: A regularized limited memory BFGS method for nonconvex unconstrained minimization. Numer. Algorithms 65(2), 305–323 (2014)
Liu, T.W., Li, D.H.: A pratical update criterion for SQP method. Optim. Methods Softw. 22(2), 253–266 (2007)
Liu, D.C., Nocedal, J.: On the limited memory BFGS method for large scale optimization. Math. Program. 45, 503–528 (1989)
Mascarenhas, W.F.: The BFGS method with exact line searches fails for non-convex objective functions. Math. Program. A 99(1), 49–61 (2004)
Nash, S.G., Nocedal, J.: A numerical study of a limited memory BFGS method and the truncated Newton method for large-scale optimization. SIAM J. Optim. 1(3), 358–372 (1991)
Nocedal, J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35, 773–782 (1980)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research, 2nd edn. Springer, Berlin (2006)
Wei, Z., Li, G., Qi, L.: New quasi-Newton methods for unconstrained optimization problems. Appl. Math. Comput. 175, 1156–1188 (2006)
Xiao, Y., Wei, Z., Wang, Z.: A limited memory BFGS-type method for large-scale unconstrained optimization. Comput. Math. Appl. 56, 1001–1009 (2008)
Xu, C.X., Zhang, J.Z.: Properties and Numerical Performance of Quasi Newton Methods with Modified Quasi-Newton Equations, Technical Report. Department of Mathematics, City University of Hong Kong (1999)
Zhang, J.Z., Deng, N.Y., Chen, L.H.: New quasi-Newton equation and related methods for unconstrained optimization. J. Optim. Theory Appl. 102, 147–167 (1999)
Zhang, J.Z., Xu, C.X.: Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations. J. Comput. Appl. Math. 137, 269–278 (2001)
Zhou, W., Zhang, L.: Global convergence of a regularized factorized quasi-Newton method for nonlinear least squares problems. Comput. Appl. Math. 29, 195–214 (2010)
Acknowledgments
The author would like to thank the research council of K.N. Toosi University of Technology and the SCOPE research center. The authors would also like to thank Prof. M. Al-Baali for his helpful comments on this work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tarzanagh, D.A., Peyghami, M.R. A new regularized limited memory BFGS-type method based on modified secant conditions for unconstrained optimization problems. J Glob Optim 63, 709–728 (2015). https://doi.org/10.1007/s10898-015-0310-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-015-0310-7