Error estimates for halfspace regularization of state constrained multiobjective elliptic control problems

Abstract

This paper generalizes the halfspace regularization given in (Jadamba et al. in Syst Control Lett 61(6):707–713, 2012) from scalar to multiobjective optimization for an abstract linearly constrained multiobjective optimization problem. We establish quantitative error estimates for the regularization error in terms of the Hausdorff distance of the efficient set of the original problem to the regularized efficient set. We apply these results to state-constrained multiobjective optimal control problems. In particular, we consider two different situations: distributed and finite-dimensional controls, and for each one, we define two different halfspace regularization schemes given in terms of a family of triangulations of the domain. This leads to two regularized problems corresponding to finitely many pointwise and integral constraints. By applying the abstract results, we get regularization error estimates for the four control problems under general conditions.