Fast bundle-level methods for unconstrained and ball-constrained convex optimization

In this paper, we study a special class of first-order methods, namely bundle-level (BL) type methods, which can utilize historical first-order information through cutting plane models to accelerate the solutions in practice. Recently, it has been shown in Lan (149(1–2):1–45, 2015) that an accelerated prox-level (APL) method and its variant, the uniform smoothing level (USL) method, have optimal iteration complexity for solving black-box and structured convex programming (CP) problems without requiring input of any smoothness information. However, these algorithms require the assumption on the boundedness of the feasible set and their efficiency relies on the solutions of two involved subproblems. Some other variants of BL methods which could handle unbounded feasible set have no iteration complexity provided. In this work we develop the fast APL (FAPL) method and fast USL (FUSL) method that can significantly improve the practical performance of the APL and USL methods in terms of both computational time and solution quality. Both FAPL and FUSL enjoy the same optimal iteration complexity as APL and USL, while the number of subproblems in each iteration is reduced from two to one, and an exact method is presented to solve the only subproblem in these algorithms. Furthermore, we introduce a generic algorithmic framework to solve unconstrained CP problems through solutions to a series of ball-constrained CP problems that also exhibits optimal iteration complexity. Our numerical results on solving some large-scale least squares problems and total variation based image reconstructions have shown advantages of these new BL type methods over APL, USL, and some other first-order methods.