MPEC Problem Formulations in Chemical Engineering Applications

Assuming that x = 0 at the solution, then the active set c(x, y) = 0 consists ofg 1 and g 3 where:∇g 1 = [1 0] T (10)∇g 3 = [y 0] T (11)[ ] 1 0∇c T = ⇒ linearly dependent (12)y 0Failure of LICQ is noteworthy as uniqueness of the constraint multipliers holds ifand only if LICQ is satisfied. A weaker condition is the Mangasarian-FromovitzConstraint Qualification (MFCQ), which requires linear independence of theequality constraints and the existence of a search direction that lies both in thenull space of the equality constraint gradients and points into the interior ofthe region defined by the linearized active inequality constraints. MFCQ is anecessary and sufficient condition for boundedness of the multipliers. However,MPCCs violate MFCQ at all feasible points as well. Consequently, multipliersof the MPCC (3) will be non-unique and unbounded.Because these constraint qualifications do not hold, it is not surprising thatfinding an MPCC solution may be difficult. In order to classify this solution, wedefine a B-stationary point for an MPCC if it is feasible to the MPCC and d = 0is the solution to the following Linear Program with Equilibrium Constraints(LPEC):wheremind∇f(w ∗ ) T d (13a)s.t. g(w ∗ ) + ∇g(w ∗ ) T d ≥ 0 (13b)h(w ∗ ) + ∇h(w ∗ ) T d = 0(13c)0 ≤ x ∗ + d x ⊥ y ∗ + d y ≥ 0 (13d)w ∗ = (x ∗ , y ∗ , z ∗ )I X = {i : x ∗ i = 0}I Y = {i : y ∗ i = 0}Verification of this condition may require the solution of 2 m linear programswhere m is the cardinality of the biactive set I X ∩ I Y . A stronger, but moretractable form of stationarity for the MPCC problem is strong stationarity. Apoint is strongly stationary if it is feasible for the MPCC and d = 0 solves the5