Ergodic Results and Bounds on the Optimal Value in Subgradient Optimization

Abstract

Subgradient methods for nonsmooth optimization have rendered considerable popularity, mainly in the context of Lagrangean relaxation; this is due to their simplicity and proven practical usefulness. We consider the minimization of a nonsmooth convex function over a convex and compact set, using a (conditional) subgradient optimization scheme with divergent series step lengths. We show that such a scheme may be augmented by a convergent lower bounding procedure based on the computation of an ergodic (averaged) sequence of subgradients and of affine functions underestimating the objective. The sequence of lower bounds enables a judgement of the quality of the solutions obtained, and can thus be used in a termination criterion. Numerical experience from an application of the bounding procedure within a subgradient scheme for a nonsmooth formulation of the linear multicommodity network flow problem is presented.