Towards simulation based mixed‐integer optimization with differential equations

We propose a decomposition based method for solving mixed‐integer nonlinear optimization problems with “black‐box” nonlinearities, where the latter, for example, may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed‐integer linear master problem and a separation problem for iteratively refining the mixed‐integer linear relaxation of the nonlinear equalities. The latter yield nonconvex feasible sets for the optimization model but we have to restrict ourselves to convex and monotone constraint functions. Under these assumptions, we prove that our algorithm finitely terminates with a global optimal solution of the mixed‐integer nonlinear problem. Additionally, we show the applicability of our approach for three applications from optimal control with integer variables, from the field of pressurized flows in pipes with elastic walls, and from steady‐state gas transport. For the latter we also present promising numerical results of our method applied to real‐world instances that particularly show the effectiveness of our method for problems defined on networks.