Proximal Nonlinear Conjugate Gradient Methods for Composite Optimization

Abstract

The nonlinear conjugate gradient methods are known to be an effective approach for standard unconstrained optimization problems especially for large-scale problems.This paper proposes a proximal nonlinear conjugate gradient method, which extends the nonlinear conjugate gradient methods to composite objective functions, namely, the sum of a smooth nonconvex function and a nonsmooth convex function, and its extension to the case where the nonsmooth function is weakly convex. The proposed method uses the \textit{forward-backward residual} which is defined by using the proximal mapping instead of the gradient and determines the search direction based on the three-term Hestenes-Stiefel (HS) formula. We establish the global convergence of the proposed algorithm under standard assumptions for both convex and weakly convex nonsmooth functions, and analyze its convergence rate in terms of stationarity.In addition, when the smooth term is strongly convex, we prove that the generated sequence converges to the optimal solution and establish its convergence rate. Finally, numerical experiments show that the proposed method is stable and achieves better performance than existing methods in both convex and nonconvex settings.