Ergodic results and bounds on the optimal value in subgradient optimization

Paper in proceeding, 1996

Subgradient methods are popular tools for nonsmooth, convex minimization, especially in the context of Lagrangean relaxation; their simplicity has been a main contribution to their success. As a consequence of the nonsmoothness, it is not straightforward to monitor the progress of a subgradient method in terms of the approximate fulfilment of optimality conditions, since the subgradients used in the method will, in general, not accumulate to subgradients that verify optimality of a solution obtained in the limit. Further, certain supplementary information, such as convergent estimates of Lagrange multipliers, is not directly available in subgradient schemes. As a means for overcoming these weaknesses of subgradient optimization methods, we introduce the computation of an ergodic (averaged) sequence of subgradients. Specifically, we consider a nonsmooth, convex program solved by a conditional subgradient optimization scheme (of which the traditional sub gradient optimization method is a special case) with divergent series step lengths, which generates a sequence of iterates that converges to an optimal solution. We show that the elements of the ergodic sequence of subgradients in the limit fulfill the optimality conditions at this optimal solution. Further, we use the convergence properties of the ergodic sequence of subgradients to establish convergence of an ergodic sequence of Lagrange multipliers. Finally, some potential applications of these ergodic results are briefly discussed.