A diagonally scaled Newton-type proximal method for minimization of the models with nonsmooth composite cost functions

Based on the memoryless BFGS (Broyden-Fletcher-Goldfarb-Shanno) updating formula of a recent well-structured diagonal approximation of the Hessian, we propose an improved proximal method for solving the minimization problem of nonsmooth composite functions. More exactly, a diagonally scaled matrix is iteratively used to approximate Hessian of the smooth ingredient of the cost function, which leads to straightly determining the search directions in each iteration. Afterward, in light of the Zhang-Hager nonmonotone scheme, a nonmonotone technique for performing the line search for the unconstrained optimization models with composite cost functions is devised. What is more, we address convergence of the suggested proximal algorithm. We close the discussion by empirically studying performance of the proposed algorithm on some large-scale compressive sensing and sparse logistic regression problems.