A Branch-and-Bound Approach to Mixed-Integer Optimal Control Using POD

Abstract

In this thesis we consider a mixed-integer optimal control problem motivated by energy-efficient building operations. The aim is to choose suitable insulation materials for outer walls from a set of integer values and determine an underfloor heating strategy to maintain a desired temperature in a room while keeping the costs reasonably low. Mathematically, this translates into a mixed-integer optimal control problem that combines two major fields, mixed-integer programming (for which we will present the branch-and-bound algorithm) and optimal control (which we will speed-up by a reduced-order model approach). The branch-and-bound algorithm performs a tree-search by iteratively solving relaxed problems (without integrality constraint, i.e. restricted linear-quadratic optimal control problems) for various lower and upper bounds on the boundary control. Consequently, many state and adjoint equations have to be solved which is time-consuming as their finite element discretizations lead to large degrees of freedom. Therefore, we apply a model reduction technique using proper orthogonal decomposition providing a significant speed-up alongside small errors as few specific ansatz functions can capture the main characteristics of the system dynamics. We discuss strategies for branching decisions and node selection in the branch-and-bound algorithm and present numerical experiments for different scenarios.