Determine the exact positive values for inactive constraints in the equality-constrained formulation

Determine the exact strictly positive setpoint values z_j > 0 corresponding to constraints g_j(u,d) that are not optimally active within the equality-constrained formulation g_i(u,d) = z_i used to solve the steady-state real-time optimization problem min_{u in U} J(u,d) subject to g_i(u,d) ≥ 0, when J(u,d) and g_i(u,d) are unknown but measured in real time.

Background

After converting inequality constraints into equality constraints with setpoints, the authors point out that for constraints that are not optimally active, the exact positive values that such constraints should take at optimum are unknown. This uncertainty is addressed in the proposed ECCBO framework by treating these values as decision variables for Bayesian optimization, thereby maintaining feasibility while optimizing the cost.