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 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 global convergence under standard assumptions, both convex and weakly convex nonsmooth fuctions. In addition, we characterize the convergence rate when the smooth term is strongly convex. Finally, numerical experiments show that the proposed method is stable and achieves better performance than existing methods in both convex and nonconvex settings.