MPEC Problem Formulations in Chemical Engineering Applications

following relaxed problem:mind∇f(w ∗ ) T d (14a)s.t. g(w ∗ ) + ∇g(w ∗ ) T d ≥ 0 (14b)h(w ∗ ) + ∇h(w ∗ ) T d = 0(14c)d x,i = 0, i ∈ I X \ I Y (14d)d y,i = 0, i ∈ I Y \ I X (14e)d x,i ≥ 0, i ∈ I X ∩ I Y (14f)d y,i ≥ 0, i ∈ I X ∩ I Y (14g)Strong stationarity implies no feasible descent direction for (14) and is equivalentto B-stationarity if the biactive set I X ∩ I Y is empty, or if the MPEC-LICQproperty holds [1]. MPEC-LICQ requires that the following set of vectors belinearly independent:{∇g i (w ∗ )|i ∈ I g } ∪ {∇h(w ∗ )} ∪ {∇x i |i ∈ I X } ∪ {∇y i |i ∈ I Y } (15)whereI g = {i : g i (w ∗ ) = 0}.MPEC-LICQ implies that the multipliers of (14) are bounded and unique. Ithas been shown that MPEC-LICQ is a generic property of MPCCs and of bilevelproblems formulated as MPCC [27] and that this property can be expected tohold “almost everywhere”. That is, if MPEC-LICQ is violated for a particularMPEC, it can always be satisfied by small perturbations of that problem.Satisfaction of MPEC-LICQ leads to the following result:Theorem 1 [1, 15, 26] If w ∗ is a solution to the MPCC (3) and MPEC-LICQholds at w ∗ , then w ∗ is strongly stationary.With this property, convergence to a strongly stationary point is much easier(and required for many NLP algorithms), since only one set of MPEC multipliersis needed to verify optimality and, consequently, this property allows thereformulation of (3) to a number of equivalent nonlinear programs. In the past,special MPEC solvers were required [4, 5, 17] to handle these difficulties. Withthis property, MPECs can now be addressed directly through the formulationand direct solution as NLPs. These NLP reformulations are covered in the nextsection.Finally, a related (but weaker and implied by MPEC-LICQ) constraint qualificationis MPEC-MFCQ, which requires that there exist a nonzero vector6