Combinatorial Integral Approximation Decompositions for Mixed-Integer Optimal Control

Abstract

Solving mixed-integer nonlinear programs (MINLPs) is hard from both a theoretical and practical perspective. Decomposing the nonlinear and the integer part is promising from a computational point of view. In general, however, no bounds on the objective value gap can be established and iterative procedures with potentially many subproblems are necessary. The situation is different for mixed-integer optimal control problems with binary variables that switch over time. Here, a priori bounds were derived for a decomposition into one continuous nonlinear control problem and one mixed-integer linear program, the combinatorial integral approximation (CIA) problem. In this article, we generalize and extend the decomposition idea. First, we derive different decompositions and analyze the implied a priori bounds. Second, we propose several strategies to recombine promising candidate solutions for the binary control functions in the original problem. We present the extensions for ordinary differential equations-constrained problems. These extensions are transferable in a straightforward way, though, to recently suggested variants for certain partial differential equations, for algebraic equations, for additional combinatorial constraints, and for discrete time problems. We implemented all algorithms and subproblems in AMPL for a proof-of-concept study. Numerical results show the improvement compared to the standard CIA decomposition with respect to objective function value and compared to general-purpose MINLP solvers with respect to runtime.