Abstract

This talk is concerned with infinite-dimensional optimization problems where a distributed function should only take on values from a set of allowed states. This property can be promoted with the aid of a L⁰-type penalty that is zero on the admissible set and one otherwise. Possible applications include sparse, integer (“multi-bang”) and switching control. Although functionals involving such binary terms are non-convex and lack weak lower-semicontinuity, application of Fenchel duality yields a formal primal-dual optimality system that admits a unique solution. This solution is in general only suboptimal, but the optimality gap can be characterized and shown to be zero under appropriate conditions. A regularized semismooth Newton method allows the numerical computation of (sub)optimal solutions. For the case of multi-bang controls, in certain situations it is possible to derive a generalized multi-bang principle, i. e., to prove that the control almost everywhere takes on allowed values except possibly on a singular set. Numerical examples illustrate the effectiveness of the proposed approach.