Nonsmooth Optimization Using Uncontrolled Inexact Information

We consider convex nonsmooth optimization problems whose objective function is known through a (fine) oracle together with some additional (cheap but poor) information – formalized as a second coarse oracle wih uncontrolled inexactness. It is the case when the objective function is itself the output of an optimization solver, using a branch-and-bound procedure, or decomposing the problem into parallel subproblems. Minimizing such objective function can be done by any bundle algorithm using only the (fine) oracle. In this paper, we propose a general scheme to incorporate the available coarse information into bundle-type methods in view of generating better iterates and then accelerating the algorithms. We study two pratical versions of the scheme: a (simple) inexact Kelley method, and a (sophisticated) level bundle method. We prove that these methods are convergent, and we present numerical illustrations showing they speed up resolution.