An accelerated subspace minimization three-term conjugate gradient algorithm for unconstrained optimization

A three-term conjugate gradient algorithm for large-scale unconstrained optimization using subspace minimizing technique is presented. In this algorithm the search directions are computed by minimizing the quadratic approximation of the objective function in a subspace spanned by the vectors: -g(k+1), s(k) and y(k) . The search direction is considered as: d (k+1) = -g (k+1) + a(k)s(k) + b(k)y(k) , where the scalars a(k) and b(k) are determined by minimization the affine quadratic approximate of the objective function. The step-lengths are determined by the Wolfe line search conditions. We prove that the search directions are descent and satisfy the Dai-Liao conjugacy condition. The suggested algorithm is of three-term conjugate gradient type, for which both the descent and the conjugacy conditions are guaranteed. It is shown that, for uniformly convex functions, the directions generated by the algorithm are bounded above, i.e. the algorithm is convergent. The numerical experiments, for a set of 750 unconstrained optimization test problems, show that this new algorithm substantially outperforms the known Hestenes and Stiefel, Dai and Liao, Dai and Yuan and Polak, Ribiere and Poliak conjugate gradient algorithms, as well as the limited memory quasi-Newton method L-BFGS and the discrete truncated-Newton method TN.