Two-step conjugate gradient method for unconstrained optimization

Using Taylor’s series, we propose a modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, using this relation and an approach introduced by Dai and Liao, we present a conjugate gradient algorithm to solve unconstrained optimization problems. The proposed method makes use of both gradient and function values, and utilizes information from the two most recent steps, while the usual secant relation uses only the latest step information. Under appropriate conditions, we show that the proposed method is globally convergent without needing convexity assumption on the objective function. Comparative results show computational efficiency of the proposed method in the sense of the Dolan–Moré performance profiles.