OPTEC - K.U.Leuven Center of Excellence: Optimization in Engineering

Direct Methods for Nonlinear Mixed--Integer Optimal Control Problems

Sebastian Sager (Univ. Heidelberg)

Many practical optimal control problems include discrete decisions. These
may be either time--independent parameters or time--dependent control
functions as gears or valves that can only take discrete values at any
given time. While great progress has been achieved in the solution of
optimization problems involving integer variables, in particular
mixed--integer linear programs, as well as in continuous optimal control
problems, the combination of the two is yet an open field of research.

We consider the question of lower bounds that can be obtained by a
relaxation of the integer requirements. For general nonlinear
mixed--integer programs such lower bounds typically suffer from a huge
integer gap. We convexify and relax the original problem and show that the
optimal solution of this continuous control problem yields the best lower
bound for the nonlinear integer problem.

Building on this theoretical result we present a novel algorithm to solve
mixed--integer optimal control problems, with a focus on discrete--valued
control functions. Our algorithm is based on the direct multiple shooting
method, an adaptive refinement of the underlying control discretization
grid and tailored integer methods. Its applicability is shown by
challenging applications, e.g., the energy optimal control of a subway
train with discrete gears and velocity limits, a batch process with time--
and tray--dependent reusage of slop (waste) cuts and port switching in
chromatographic separation processes.