Author(s)

Tsegay Giday Woldu¹, Haibin Zhang¹, Yemane Hailu Fissuh²

¹Department of Applied Mathematics, Beijing University of Technology, Beijing, China.
²Department of Statistics, Aksum University, Aksum, Ethiopia.

In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmonotone line search technique. The method incorporates the modified BFGS secant equation in an effort to include the second order information of the objective function. The new secant equation has both gradient and function value information, and its update formula inherits the positive definiteness of Hessian approximation for general convex function. In order to improve the likelihood of finding a global optimal solution, we introduce a new modified nonmonotone line search technique. It is shown that, for nonsmooth convex problems, the proposed algorithm is globally convergent. Numerical results show that this new scaled conjugate gradient algorithm is promising and efficient for solving not only convex but also some large scale nonsmooth nonconvex problems in the sense of the Dolan-Moré performance profiles.

Conjugate Gradient Method, BFGS Method, Modified Secant Equation, Nonmonotone Line Search, Nonsmooth Optimization

Woldu, T. , Zhang, H. and Fissuh, Y. (2020) A Scaled Conjugate Gradient Method Based on New BFGS Secant Equation with Modified Nonmonotone Line Search. American Journal of Computational Mathematics, 10, 1-22. doi: 10.4236/ajcm.2020.101001.