A Novel Quasi-Newton Method for Composite Convex Minimization

•A novel parallelizable Jacobi iteration type non-smooth convex composite optimization method is proposed.•Both first and second order information are utilised while the typical state-of-the-art methods only use first order information.•The state of art second-order technique, BFGS quasi-Newton method, as well as 3 first-order techniques, steepest gradient descent for L2 and L1-norms and Nesterov's accelerated gradient descent, are integrated into the proposed method.•A convergence rate with a lower bound of O(1k2) and superlinear convergence is enjoyable.•The proposed method converge significantly superior to the state of art methods (e.g. APA-APG1 and APA-APG2, methods which enjoy O(1k) convergence)