How Important is the Detection of Changes in Active Constraints in Real-Time Optimization?

In real-time optimization, enforcing the constraints that need to be active is important for optimality. In fact, it has been established in the context of parametric variations that, if these constraints are not satisfied, the optimality loss would be O($\eta^2$) – denoting the magnitude of the parametric variations. In contrast, the loss of optimality upon enforcing the correct set of active constraints would be O($\eta^2$). However, no result is available when the set of active constraints changes due to parametric variations, which forms the subject of this paper. Herein it is shown that, if the optimal solution is unique for each , keeping only the strictly active constraints of the nominal solution active will lead to O($\eta^2$) loss in optimality, even when the remaining active constraints of the perturbed system are different from that of the nominal system. This, in turn, means that, in any input adaptation scheme for real-time optimization, identifying changes in active constraints is not important as long as it is possible to enforce the strictly active constraints of the nominal solution to remain active.