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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.].
MSC:
90C53 Methods of quasi-Newton type
49M37 Numerical methods based on nonlinear programming
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
Software:
CUTEr; SCALCG
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References:
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