Bundle-Level Type Methods Uniformly Optimal for Smooth and Nonsmooth Convex Optimization

TL;DR

This paper introduces new uniformly optimal first-order methods for convex programming that do not require problem parameters and adapt to smooth or nonsmooth cases, achieving optimal complexity.

Contribution

It develops an accelerated bundle-level method and a practical restricted memory version, extending to composite and saddle-point problems without needing smoothness information.

Findings

Achieves optimal iteration complexity for general convex problems.

Effective for semidefinite and stochastic programming.

Does not require prior knowledge of problem smoothness.

Abstract

The main goal of this paper is to develop uniformly optimal first-order methods for convex programming (CP). By uniform optimality we mean that the first-order methods themselves do not require the input of any problem parameters, but can still achieve the best possible iteration complexity bounds. By incorporating a multi-step acceleration scheme into the well-known bundle-level method, we develop an accelerated bundle-level (ABL) method, and show that it can achieve the optimal complexity for solving a general class of black-box CP problems without requiring the input of any smoothness information, such as, whether the problem is smooth, nonsmooth or weakly smooth, as well as the specific values of Lipschitz constant and smoothness level. We then develop a more practical, restricted memory version of this method, namely the accelerated prox-level (APL) method. We investigate the generalization of the APL method for solving certain composite CP problems and an important class of saddle-point problems recently studied by Nesterov [Mathematical Programming, 103 (2005), pp 127-152]. We present promising numerical results for these new bundle-level methods applied to solve certain classes of semidefinite programming (SDP) and stochastic programming (SP) problems.