A Class of Derivative-Free CG Projection Methods for Nonsmooth Equations with an Application to the LASSO Problem

Abstract

In this paper, based on a modified Gram–Schmidt (MGS) process, we propose a class of derivative-free conjugate gradient (CG) projection methods for nonsmooth equations with convex constraints. Two attractive features of the new class of methods are (1) its generated direction contains a free vector, which can be set as any vector such that the denominator of the direction does not equal to zero; (2) it adopts a new line search which can reduce its computing cost. The new class of methods includes many efficient iterative methods for the studied problem as its special cases. When the underlying mapping is monotone, we establish its global convergence and convergence rate. Finally, preliminary numerical results about the LASSO problem show that the new class of methods is promising compared to some existing ones.