Direct methods with maximal lower bound for mixed-integer optimal control problems |
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Authors: | Sebastian Sager Hans Georg Bock Gerhard Reinelt |
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Institution: | (1) Interdisciplinary Center for Scientific Computing, INF 368, 69120 Heidelberg, Germany |
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Abstract: | 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 (with respect to binary controls) and relax the original
problem and prove 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 heuristic integer methods. Its applicability is shown
by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.
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Keywords: | Optimal control Mixed-integer programming Hybrid systems |
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