An adaptive nonmonotone global Barzilai–Borwein gradient method for unconstrained optimization

Abstract

The Barzilai–Borwein (BB) gradient method has received many studies due to its simplicity and numerical efficiency. By incorporating a nonmonotone line search, Raydan (SIAM J Optim. 1997;7:26–33) has successfully extended the BB gradient method for solving general unconstrained optimization problems so that it is competitive with conjugate gradient methods. However, the numerical results reported by Raydan are poor for very ill-conditioned problems because the effect of the degree of nonmonotonicity may be noticeable. In this paper, we focus more on the nonmonotone line search technique used in the global Barzilai–Borwein (GBB) gradient method. We improve the performance of the GBB gradient method by proposing an adaptive nonmonotone line search based on the morphology of the objective function. We also prove the global convergence and the R-linear convergence rate of the proposed method under reasonable assumptions. Finally, we give some numerical experiments made on a set of unconstrained optimization test problems of the CUTEr collection. The results show the efficiency of the proposed method in the sense of the performance profile introduced (Math Program. 2002;91:201–213) by Dolan and Moré.