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Global convergence of a modified Fletcher-Reeves conjugate gradient method with Wolfe line search. (English) Zbl 1464.65058

Summary: In this paper, we are concerned with the conjugate gradient methods for solving unconstrained optimization problems. We propose a modified Fletcher-Reeves (abbreviated FR) [R. Fletcher and C. M. Reeves, Comput. J. 7, 149–154 (1964; Zbl 0132.11701)] conjugate gradient algorithm satisfying a parametrized sufficient descent condition with a parameter \(\delta_k\) is proposed. The parameter \(\delta_k\) is computed by means of the conjugacy condition, thus an algorithm which is a positive multiplicative modification of the Hestenes and Stiefel (abbreviated HS) [M. R. Hestenes and E. Stiefel, J. Res. Natl. Bur. Stand. 49, 409–436 (1952; Zbl 0048.09901)] algorithm is obtained, which produces a descent search direction at every iteration that the line search satisfies the Wolfe conditions. Under appropriate conditions, we show that the modified FR method with the strong Wolfe line search is globally convergent of uniformly convex functions. We also present extensive preliminary numerical experiments to show the efficiency of the proposed method.

MSC:

65K05 Numerical mathematical programming methods
90C25 Convex programming
90C26 Nonconvex programming, global optimization
90C27 Combinatorial optimization
90C30 Nonlinear programming

Software:

CUTE; SCALCG; CUTEr
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Full Text: DOI

References:

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