Identify the optimally active constraints in the unknown real-time optimization problem

Identify the subset of constraints g_i(u,d) that are optimally active at the steady-state optimum of the real-time optimization problem min_{u in U} J(u,d) subject to g_i(u,d) ≥ 0, where the cost J(u,d) and constraints g_i(u,d) are unknown but measured in real time.

Background

The paper considers steady-state real-time optimization with unknown cost and constraints, relying on real-time measurements rather than detailed process models. In transforming the inequality-constrained problem to an equality-constrained form using constraint setpoints, the authors note a core challenge: the set of constraints that are optimally active at the solution is not known a priori. This uncertainty motivates their embedded constraint-control strategy and the use of Bayesian optimization to search over constraint setpoints, ensuring feasibility while optimizing performance.