zbMATH — the first resource for mathematics

A modified scaled memoryless symmetric rank-one method. (English) Zbl 1448.90100
Summary: To guarantee heredity of positive definiteness under the popular Wolfe line search conditions, a modification is made on the symmetric rank-one updating formula, a simple quasi-Newton approximation for (inverse) Hessian of the objective function of an unconstrained optimization problem. Then, the scaling approach is employed on a memoryless version of the proposed formula, leading to an iterative method which is appropriate for solving large-scale problems. Based on an eigenvalue analysis, it is shown that the self-scaling parameter proposed by S. S. Oren and E. Spedicato [Math. Program. 10, 70–90 (1976; Zbl 0342.90045)] is an optimal parameter for the proposed updating formula in the sense of minimizing the condition number. Also, a sufficient descent property is established for the method, together with a global convergence analysis for uniformly convex objective functions. Numerical experiments demonstrate computational efficiency of the proposed method with the self-scaling parameter proposed in [loc. cit.].
90C53 Methods of quasi-Newton type
49M37 Numerical methods based on nonlinear programming
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
Full Text: DOI
[1] Andrei, N., Accelerated scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Eur. J. Oper. Res., 204, 3, 410-420 (2010) · Zbl 1189.90151
[2] Arguillère, S., Approximation of sequences of symmetric matrices with the symmetric rank-one algorithm and applications, SIAM J. Matrix Anal. Appl., 36, 1, 329-347 (2015) · Zbl 1314.90089
[3] Babaie-Kafaki, S., A quadratic hybridization of Polak-Ribière-Polyak and Fletcher-Reeves conjugate gradient methods, J. Optim. Theory Appl., 154, 3, 916-932 (2012) · Zbl 1256.90050
[4] Babaie-Kafaki, S., On optimality of the parameters of self-scaling memoryless quasi-Newton updating formulae, J. Optim. Theory Appl., 167, 91-101 (2015) · Zbl 1327.90394
[5] Babaie-Kafaki, S.; Ghanbari, R.; Mahdavi-Amiri, N., Two new conjugate gradient methods based on modified secant equations, J. Comput. Appl. Math., 234, 5, 1374-1386 (2010) · Zbl 1202.65071
[6] Barzilai, J.; Borwein, JM, Two-point stepsize gradient methods, IMA J. Numer. Anal., 8, 1, 141-148 (1988) · Zbl 0638.65055
[7] Conn, AR; Gould, NIM; Toint, PhL, Convergence of quasi-Newton matrices generated by the symmetric rank-one update, Math. Program., 50, 2, 177-195 (1991) · Zbl 0737.90062
[8] Dai, YH; Han, JY; Liu, GH; Sun, DF; Yin, HX; Yuan, YX, Convergence properties of nonlinear conjugate gradient methods, SIAM J. Optim., 10, 2, 348-358 (1999)
[9] Dolan, ED; Moré, JJ, Benchmarking optimization software with performance profiles, Math. Program., 91, 2, 201-213 (2002) · Zbl 1049.90004
[10] Gould, NIM; Orban, D.; Toint, PhL, CUTEr: a constrained and unconstrained testing environment, revisited, ACM Trans. Math. Softw., 29, 4, 373-394 (2003) · Zbl 1068.90526
[11] Nocedal, J.; Wright, SJ, Numerical Optimization (2006), New York: Springer, New York
[12] Oren, SS, Self-scaling variable metric (SSVM) algorithms. II. Implementation and experiments, Manag. Sci., 20, 5, 863-874 (1974) · Zbl 0316.90065
[13] Oren, S.S., Luenberger, D.G.: Self-scaling variable metric (SSVM) algorithms. I. Criteria and sufficient conditions for scaling a class of algorithms. Manag. Sci., 20(5), 845-862, (1973/74) · Zbl 0316.90064
[14] Oren, SS; Spedicato, E., Optimal conditioning of self-scaling variable metric algorithms, Math. Program., 10, 1, 70-90 (1976) · Zbl 0342.90045
[15] Sugiki, K.; Narushima, Y.; Yabe, H., Globally convergent three-term conjugate gradient methods that use secant conditions and generate descent search directions for unconstrained optimization, J. Optim. Theory Appl., 153, 3, 733-757 (2012) · Zbl 1262.90170
[16] Sun, W.; Yuan, YX, Optimization Theory and Methods: Nonlinear Programming (2006), New York: Springer, New York
[17] Watkins, DS, Fundamentals of Matrix Computations (2002), New York: John Wiley and Sons, New York · Zbl 1005.65027
[18] Xu, C.; Zhang, JZ, A survey of quasi-Newton equations and quasi-Newton methods for optimization, Ann. Oper. Res., 103, 1-4, 213-234 (2001) · Zbl 1007.90069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.