A Generalized Alternating Linearization Bundle Method for Structured Convex Optimization with Inexact First-Order Oracles

Abstract

In this paper, we consider a class of structured optimization problems whose objective function is the summation of two convex functions: f and h, which are not necessarily differentiable. We focus particularly on the case where the function f is general and its exact first-order information (function value and subgradient) may be difficult to obtain, while the function h is relatively simple. We propose a generalized alternating linearization bundle method for solving this class of problems, which can handle inexact first-order information of on-demand accuracy. The inexact information can be very general, which covers various oracles, such as inexact, partially inexact and asymptotically exact oracles, and so forth. At each iteration, the algorithm solves two interrelated subproblems: one aims to find the proximal point of the polyhedron model of f plus the linearization of h; the other aims to find the proximal point of the linearization of f plus h. We establish global convergence of the algorithm under different types of inexactness. Finally, some preliminary numerical results on a set of two-stage stochastic linear programming problems show that our method is very encouraging.