A Non-monotone Line Search Algorithm for Unconstrained Optimization

The monotone line search schemes have been extensively used in the iterative methods for solving various optimization problems. It is well known that the non-monotone line search technique can improve the likelihood of finding a global optimal solution and the numerical performance of the methods, especially for some difficult nonlinear problems. The traditional non-monotone line search approach requires that a maximum of recent function values decreases. In this paper, we propose a new line search scheme which requires that a convex combination of recent function values decreases. We apply the new line search technique to solve unconstrained optimization problems, and show the proposed algorithm possesses global convergence and R-linear convergence under suitable assumptions. We also report the numerical results of the proposed algorithm for solving almost all the unconstrained testing problems given in CUTEr, and give numerical comparisons of the proposed algorithm with two famous non-monotone methods.