When nonsmooth, convex minimization problems are solved by subgradient optimization methods, the subgradients used will in generalnotaccumulate to subgradients which verify the optimality of a solution obtained in the limit. It is therefore not a straightforward task to monitor the progress of a subgradient method in terms of the approximate fulfillment of optimality conditions. Further, certain supplementary information, such as convergent estimates of Lagrange multipliers and convergent lower bounds on the optimal objective value, is not directly available in subgradient schemes