Advanced Optimization for Process Systems Engineering

This chapter addresses the problem of establishing the feasibilty of a set of constraints given that recourse variables are involved, and that the uncertainty set is specified, typically through lower and upper bounds. This problem, denoted as the feasibility test problem, is shown to correspond to a max-min-max optimization problem. It is shown that, under assumptions of convexity, the problem can be simplified through vertex seaches in the parameter set. It is also shown that the feasibility test problem can be reformulated asa bilevel optimization problem in which the KKT conditions in the inner problem can be reformulated through mixed-integer constraints. It is shown that this MINLP has the capability of predicting nonvertex solutions. The feasibility test is then extended to the feasibility index problem that determines the actual parameter range that is feasible. The concept of one-dimensional convexity is introduced to provide sufficient conditions for the validity of vertex searches. The example of a heat exchanger network is used to illustrate the mathematical formulations.