Optimization over decision trees: a case study for the design of stable direct-current electricity networks

In many real-world mixed-integer optimization problems from engineering, the side constraints can be subdivided into two categories: constraints which describe a certain logic to model a feasible allocation of resources (such as a maximal number of available assets, working time requirements, maintenance requirements, contractual obligations, etc.), and constraints which model physical processes and the related quantities (such as current, pressure, temperature, etc.). While the first type of constraints can often easily be stated in terms of a mixed-integer program (MIP), the second part may involve the incorporation of complex non-linearities, partial differential equations or even a black-box simulation of the involved physical process. In this work, we propose the integration of a trained tree-based classifier - a decision-tree or a random forest, into a mixed-integer optimization model as a possible remedy. We assume that the classifier has been trained on data points produced by a detailed simulation of a given complex process to represent the functional relationship between the involved physical quantities. We then derive MIP-representable reformulations of the trained classifier such that the resulting model can be solved using state-of-the-art solvers. At the hand of several use cases in terms of possible optimization goals, we show the broad applicability of our framework that is easily extendable to other tasks beyond engineering. In a detailed real-world computational study for the design of stable direct-current power networks, we demonstrate that our approach yields high-quality solutions in reasonable computation times.