A Method of Centers with Approximate Subgradient Linearizations for Nonsmooth Convex Optimization

We give a proximal bundle method for constrained convex optimization. It requires only evaluating the problem functions and their subgradients with an unknown accuracy ε. Employing a combination of the classic method of centers' improvement function with an exact penalty function, it does not need a feasible starting point. It asymptotically finds points with at least ε-optimal objective values that are ε-feasible. When applied to the solution of linear programming problems via column generation, it allows for ε-accurate solutions of column generation subproblems.