Black-box cutting plane (BBCP): a framework for hybrid black-box/glass-box mixed-integer optimization

Hybrid black-box/glass-box optimization deals with problems where some functions lack explicit expressions, making traditional model-based optimization unsuitable. Cutting plane methods are promising for such problems, however, cut generation in black-box settings is non-trivial due to the absence of gradient information. This work introduces a black-box cutting plane (BBCP) framework for mixed-integer constrained problems with one or several black-box terms in the objective. BBCP relies on a decomposition of the problem to detach glass-box and black-box structures. At each iteration, BBCP samples multiple candidates to generate cuts. Cut generation strategies based on generalized simplex gradients and auxiliary linear programming models are proposed. BBCP brings novelty by offering flexibility and allowing users to experiment with different cut generation strategies while ensuring finite-time termination under standard assumptions, e.g., boundedness of the objective function and search space. Particularly, this work highlights how tailored cuts that rely on problem-specific knowledge can efficiently deal with failed black-box evaluations, i.e., hidden constraints. This adaptability makes BBCP a powerful tool for tackling mixed-integer hybrid optimization problems efficiently. The use and performance of BBCP is illustrated with the optimal design of a distillation column and the simultaneous scheduling and control of network bath processes, which highlight the importance of relying on problem knowledge to manage hidden constraints.