Globally Convergent Three-Term Conjugate Gradient Methods that Use Secant Conditions and Generate Descent Search Directions for Unconstrained Optimization

In this paper, we propose a three-term conjugate gradient method based on secant conditions for unconstrained optimization problems. Specifically, we apply the idea of Dai and Liao (in Appl. Math. Optim. 43: 87–101, 2001) to the three-term conjugate gradient method proposed by Narushima et al. (in SIAM J. Optim. 21: 212–230, 2011). Moreover, we derive a special-purpose three-term conjugate gradient method for a problem, whose objective function has a special structure, and apply it to nonlinear least squares problems. We prove the global convergence properties of the proposed methods. Finally, some numerical results are given to show the performance of our methods.