Optimizing Partially Defined Black-Box Functions Under Unknown Constraints via Sequential Model Based Optimization: An Application to Pump Scheduling Optimization in Water Distribution Networks

This paper proposes a Sequential Model Based Optimization framework for solving optimization problems characterized by a black-box, multi-extremal, expensive and partially defined objective function, under unknown constraints. This is a typical setting for simulation-optimization problems, where the objective function cannot be computed for some configurations of the decision/control variables due to the violation of some (unknown) constraint. The framework is organized in two consecutive phases, the first uses a Support Vector Machine classifier to approximate the boundary of the unknown feasible region within the search space, the second uses Bayesian Optimization to find a globally optimal (feasible) solution. A relevant difference with traditional Bayesian Optimization is that the optimization process is performed on the estimated feasibility region, only, instead of the entire search space. Some results on three 2D test functions and a real case study for the Pump Scheduling Optimization in Water Distribution Networks are reported. The proposed framework proved to be more effective and efficient than Bayesian Optimization approaches using a penalty for function evaluations outside the feasible region.