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This paper proposes a new class of accelerated conjugate-gradient-like algorithms for solving large scale unconstrained optimization problems, which combine the idea of accelerated adaptive Perry conjugate gradient algorithms proposed by Andrei (2017) with the modified secant condition and the nonmonotone line search technique. An attractive property of the proposed methods is that the search direction always provides sufficient descent step which is independent of the line search used and the convexity of objective function. Under common assumptions, it is proven that the proposed methods possess global convergence for nonconvex smooth functions, and R-linear convergence for uniformly convex functions. The numerical experiments show the efficiency of the proposed method in practical computations.

Memoryless quasi-Newton methods are studied for solving large-scale unconstrained optimization problems. Recently, memoryless quasi-Newton methods based on several kinds of updating formulas were proposed. Since the methods closely related to the conjugate gradient method, the methods are promising. In this paper, we propose a memoryless quasi-Newton method based on the Broyden family with the spectral-scaling secant condition. We focus on the convex and preconvex classes of the Broyden family, and we show that the proposed method satisfies the sufficient descent condition and converges globally. Finally, some numerical experiments are given.

In this paper, we present a new nonmonotone memory gradient algorithm for unconstrained optimization problems. An attractive property of the proposed method is that the search direction always provides sufficient descent step at each iteration. This property is independent of the line search used. Under mild assumptions, the global and local convergence results of the proposed algorithm are established respectively. Numerical results are also reported to show that the proposed method is suitable to solve large-scale optimization problems and is more stable than other similar methods in practical computation.