MPEC Problem Formulations in Chemical Engineering Applications

etter understanding of NLP-based solution strategies for MPCCs, it is essentialto consider the formulation of well-posed complementarity models. In thissection, we consider a systematic MPCC modeling strategy and apply it to anumber of widely applied process examples.To develop these concepts, we again caution that complementarity formulationsare nonconvex problems and may lead to multiple local solutions. Consequently,with the application of standard NLP solvers, the user will need tobe satisfied with local solutions, unless additional problem-specific informationcan be applied. Despite the nonconvexity in all MPCC formulations, it is clearthat poorly posed MPCC formulations also need to be avoided. For instance,the feasible region for:0 ≤ y ⊥ (1 − y) ≥ 0 (25)consists of only two isolated points, and the disjoint feasible region associatedwith this example often leads to convergence failures, unless the model is initializedclose to a solution with either y = 0 or y = 1. Moreover, without carefulconsideration, it is not difficult to create MPCCs with similar difficulties.Because the MPCC can be derived from an associated MPEC, we considerthe formulation in (2). To avoid disjoint regions, we require that both θ(x, y)and C(x) be convex in y for all (x, y) ∈ Z. This leads to a well-defined solutionfor the inner problem. Moreover, the resulting complementarities are then“strongly monotone”, which have been shown to be very useful in practice. Thisobservation leads to the following guidelines for the examples in this study:• Start with the inner level problem:minyϕ(x)y s.t. y a ≤ y ≤ y b (26)where ϕ(x) is a switching function for the discrete decision.• For (26) it is especially important that the upper level constraints in (2),i.e., (x, y) ∈ Z, not limit the selection of any value of y ∈ C(x).• The resulting MPEC is then converted to an MPCC. For instance, applyingthe KKT conditions to (26) we obtain the complementaritiesϕ(x) − s a + s b = 0 (27a)0 ≤ (y − y a ) ⊥ s a ≥ 0 (27b)0 ≤ (y b − y) ⊥ s b ≥ 0 (27c)• The relations in (27) can be further simplified through variable eliminationand application of the complementarity conditions.• Finally, the resulting complementarity conditions are incorporated withinNLPs using the formulations described in section 3.14