In the present paper, a bicriterial optimal control problem governed by an abstract evolution problem and bilateral control constraints is considered. To compute Pareto optimal points and the Pareto front numerically, the (Euclidean) reference point method is applied, where many scalar constrained optimization problems have to be solved. For this reason a reduced-order approach based on proper orthogonal decomposition (POD) is utilized. An a-posteriori error analysis ensures a desired accuracy for the Pareto optimal points and for the Pareto front computed by the POD method. Numerical experiments for evolution problems with convection-diffusion illustrate the efficiency of the presented approach.

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In the present paper a bicriterial optimal control problem governed by a parabolic partial differential equation (PDE) and bilateral control constraints is considered. For the numerical optimization the reference point method is utilized. The PDE is discretized by a Galerkin approximation utilizing the method of proper orthogonal decomposition (POD). POD is a powerful approach to derive reduced-order approximations for evolution problems. Numerical examples illustrate the efficiency of the proposed strategy.

A framework for set-oriented multiobjective optimal control of partial differential equations using reduced order modeling has recently been developed [1]. Following concepts from localized reduced bases methods, error estimators for the reduced cost functionals are utilized to construct a library of locally valid reduced order models. This way, a superset of the Pareto set can efficiently be computed while maintaining a prescribed error bound. In this article, this algorithm is applied to a problem with non-smooth objective functionals. Using an academic example, we show that the extension to non-smooth problems can be realized in a straightforward manner. We then discuss the implications on the numerical results.

In the present paper an optimal control problem governed by the heat equation is considered, where continuous as well as discrete controls are involved. To obtain the discrete controls the branch-and-bound method is utilized, where in each node a relaxed control constrained optimal control problem has to be solved involving only continuous controls. However, the solutions to many relaxed optimal control problems have to be computed numerically. For that reason model-order reduction is applied to speed-up the branch-and-bound method. In this work the method of proper orthogonal decomposition (POD) is used. A posteriori error estimation in each node ensures that the calculated solutions are sufficiently accurate. Numerical experiments illustrate the efficiency of the proposed strategy.