A Proximity Control Algorithm to Minimize Nonsmooth and Nonconvex Functions

We present a new proximity control bundle algorithm to minimize nonsmooth and nonconvex locally Lipschitz functions. In contrast with the traditional oracle-based methods in nonsmooth programming, our method is model-based and can accommodate cases where several Clarke subgradients can be computed at reasonable cost. We propose a new way to manage the proximity control parameter, which allows us to handle nonconvex objectives. We prove global convergence of our method in the sense that every accumulation point of the sequence of serious steps is critical. Our method is tested on a variety of examples in H∞-controller synthesis.