A Modified Nonlinear Conjugate Gradient Algorithm for Large-Scale Nonsmooth Convex Optimization

Abstract

Nonlinear conjugate gradient methods are among the most preferable and effortless methods to solve smooth optimization problems. Due to their clarity and low memory requirements, they are more desirable for solving large-scale smooth problems. Conjugate gradient methods make use of gradient and the previous direction information to determine the next search direction, and they require no numerical linear algebra. However, the utility of nonlinear conjugate gradient methods has not been widely employed in solving nonsmooth optimization problems. In this paper, a modified nonlinear conjugate gradient method, which achieves the global convergence property and numerical efficiency, is proposed to solve large-scale nonsmooth convex problems. The new method owns the search direction, which generates sufficient descent property and belongs to a trust region. Under some suitable conditions, the global convergence of the proposed algorithm is analyzed for nonsmooth convex problems. The numerical efficiency of the proposed algorithm is tested and compared with some existing methods on some large-scale nonsmooth academic test problems. The numerical results show that the new algorithm has a very good performance in solving large-scale nonsmooth problems.