We present a generalized version of the well-known Frank-Wolfe method for minimizing a composite functional j(u)=f(u)+g(u). Here, f is a smooth function and g is in general nonsmooth but convex. The optimization space is given as the dual space of a separable, generally non-reflexive, Banach space . We address theoretical properties of the method, such as worst-case convergence rates and mesh-independence results. For specific instances of the problem we further show how additional acceleration steps eventually lead to optimal convergence rates.
joint work with Michael Ulbrich
Motivated by optimal control problems for elliptic variational inequalities we develop an inexact bundle method for nonsmooth nonconvex optimization subject to general convex constraints. The proposed method requires only inexact evaluations of the cost function and of an element of Clarke’s subdifferential. A global convergence theory in a suitable infinite-dimensional Hilbert space setting is presented. We discuss the application of our framework to optimal control of the (stochastic) obstacle problem and present numerical results.
joint work with Boris Vexler, Philip Trautmann
We will consider a control problem (P) for the wave equation with a standard tracking-type cost functional and a semi-norm in the space of functions with bounded variations (BV) in time. The considered controls in BV act as a forcing function on the wave equation. This control problem is interesting for practical applications because the semi-norm enhances optimal controls which are piecewise constant. In this talk we focus on a variationally discretized version of (P). Under specific assumptions we can present optimal convergence rates for the controls, states, and cost functionals.