Another Conjugate Gradient Algorithm with Guaranteed Descent and Conjugacy Conditions for Large-scale Unconstrained Optimization

In this paper we suggest another accelerated conjugate gradient algorithm that for all both the descent and the conjugacy conditions are guaranteed. The search direction is selected as where , The coefficients 0 k ≥ 1 1 1 1 ( / ) ( / ) T T T T k k k k k k k k k k k k k k d g y g y s s t s g y s θ + + + + = − + − , s 1 1 ( ) k k g f x + + = ∇ 1 . k k k s x x + = − k θ and in this linear combination are selected in such a way that both the descent and the conjugacy condition are satisfied at every iteration. The algorithm introduces the modified Wolfe line search in which the parameter in the second Wolfe condition is modified at every iteration. It is shown that both for uniformly convex functions and for general nonlinear functions the algorithm with strong Wolfe line search generates directions bounded away from infinity. The algorithm uses an acceleration scheme modifying the steplength k t k α in such a manner as to improve the reduction of the function values along the iterations. Numerical comparisons with some conjugate gradient algorithms using a set of 75 unconstrained optimization problems with different dimensions, some of them from the CUTE library, show that the computational scheme outperform the known conjugate gradient algorithms like Hestenes and Stiefel; Polak, Ribière and Polyak; Dai and Yuan or the hybrid Dai and Yuan; CG_DESCENT with Wolfe line search by Hager and Zhang, as well as the quasiNewton L-BFGS by Liu and Nocedal.