A Bundle Trust-Region Method for Marginal Functions Using Continuous Outer Subdifferentials

Martin Knossalla

The theory of subdifferentials provides adequate methods and tools to put descent methods for nonsmooth optimization problems into practice. But in applications often exact information on the whole subdifferential is not available as, e.g. for marginal functions in parametric mathematical programming. In this situation the semismoothness of the objective function cannot be proved or is violated.\\
Based on continuous outer subdifferentials (COS) developed earlier, we present a constructive strategy for optimization problems with locally Lipschitz continuous objective function. At first this approach is described from a theoretical point of view and for arbitrary locally Lipschitz continuous functions. It is realized by projections onto the value of a COS. In practice, it may be that the computation of the whole COS is too difficult. For this reason we will construct and, further, approximate a COS for marginal functions. Based on this approximation a bundle trust-region method will is presented for which global convergence can be proved.