MPEC Problem Formulations in Chemical Engineering Applications

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RegEq(ǫ): min f(w) (19a)s.t. h(w) = 0 (19b)g(w) ≥ 0(19c)x, y ≥ 0 (19d)x i y i = ǫ ∀i (19e)PF(ρ): min f(w) + ρx T y (20a)s.t. h(w) = 0 (20b)g(w) ≥ 0(20c)x, y ≥ 0 (20d)Complementarity conditions may be relaxed and the problem reformulated asin the Reg(ǫ), RegComp(ǫ), or RegEq(ǫ) with a positive relaxation parameterǫ. The solution of the MPCC, w ∗ , can be obtained by solving a series of relaxedsolutions, w(ǫ), as ǫ approaches zero. These solution strategies are generally attractedto strongly stationary points, but with some exceptions (see [1]). Underassumptions of MPEC-MFCQ and sufficient second order MPEC conditions [24],Reg(ǫ) and RegComp(ǫ) exhibit local convergence of ‖w(ǫ) − w ∗ ‖ ≤ O ( ǫ 1/2) .(These properties can be strengthened to O (ǫ) if we can assume strict complementarityof the bound multipliers for i ∈ I X ∪ I Y .) Reg(ǫ) has been implementedand tested in a version of IPOPT called IPOPT-C [22]. On the otherhand, RegEq(ǫ) will exhibit slower convergence with ‖w(ǫ) − w ∗ ‖ ≤ O ( ǫ 1/4)under the same assumptions.In a related development, Nonlinear Complementarity Problem (NCP) functionsand smoothing functions have also been used extensively to solve MPCCs[3, 10, 14, 28]. NCP functions replace the complementarity condition in theproblem with an equivalent nonlinear equation. One widely studied NCP functionis the Fischer-Burmeister function:φ(x, y) = x + y − √ x 2 + y 2 (21)As this function is non-differentiable at x = y = 0, it is usually smoothed to theform:φ(x, y) = x + y − √ x 2 + y 2 + ǫ (22)for some small ǫ > 0. The solution to the original problem is then recovered bysolving a series of problems as ǫ approaches zero. An equivalence can be madebetween this method and RegEq(ǫ). Accordingly, the problem will converge atthe same slow convergence rate.In contrast to the regularized formulations, we also consider the exact l 1penalization shown in PF(ρ). Here the complementarity can be moved fromthe constraints to the objective function and the resulting problem is solvedfor a particular value of ρ. If ρ ≥ ρ c , where ρ c is the critical value of thepenalty parameter, then the complementarity constraints will be satisfied at8
the solution. Similarly, Anitescu et al. [1] consider a related “elastic mode”formulation, where artificial variables are introduced to relax the constraints inPF(ρ) and an additional l ∞ constraint penalty term is added. In both cases theresulting NLP formulation has the following properties:Theorem 2 [1] If w ∗ is a solution to PF(ρ) and w ∗ is feasible for the MPCC(3), then w ∗ is a strongly stationary solution to (3).However, the value of ρ c is not known beforehand. If the initial value is toosmall a series of problems with increasing ρ values may need to be solved, andhere we can apply the following result:Theorem 3 [15] If w ∗ is a limit point of a sequence of solutions {w k } toPF(ρ k ), MPEC-LICQ holds and ρ k x k i → 0 and ρ k yi k → 0 for i ∈ I X ∪ I Y ,then w ∗ is a strongly stationary solution to (3).Hence, the solution strategy is attracted to strongly stationary points, andwith a large enough ρ this l 1 penalization is exact in the sense that all stronglystationary points of the MPCC are local minimizers to PF(ρ). PF(ρ) allowsany general nonlinear programming solver to be used to solve a complementarityproblem. Provided that the penalty parameter is large enough, the MPCCmay then be solved as a single problem, instead of a series of problems. Finally,in order to provide greater numerical efficiency, it has been suggested todynamically change the penalty parameter during the course of the optimizationfor both active set and interior point optimization algorithms [1, 15]. Thismodification was recently introduced into the solver KNITRO [16].Finally, there is a strong interaction between the MPCC reformulation andthe applied NLP solver. In particular, as noted in [15], if a barrier NLP method(like KNITRO or IPOPT) is applied to PF(ρ), then there is a one-to-one correspondenceof intermediate solutions of PF(ρ), Reg(ǫ) and RegComp(ǫ). Thisequivalence can be seen by comparing the penalty parameter ρ to the multiplierassociated with the relaxed complementarity constraints in Reg(ǫ) andRegComp(ǫ). As a result, for a corresponding set of parameters ρ and ǫ wewould expect a barrier NLP method to exhibit similar performance with eitherPF(ρ), Reg(ǫ) or RegComp(ǫ).4 MPECLib ResultsThe formulations of the previous section provide a straightforward way to representand address any MPCC through nonlinear programming. An importantcomponent of this task is the efficient and reliability of the NLP solver. In thissection we compare and evaluate each of these formulations through a librarytest set provided with the GAMS modeling environment [8]. For this evaluationwe consider MPECLib, a large collection of test problems, and apply an automaticreformulation of MPECs provided by the NLPEC meta-solver in GAMS.9
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Page 10 and 11: 4.1 Automatic Reformulation of Comp
Page 12 and 13: 100Performance Profile9080Percent O
Page 14 and 15: etter understanding of NLP-based so
Page 16 and 17: Simplifying the complementarity con
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Page 20 and 21: 6 Case StudiesFrom the results of S
Page 22 and 23: Objective SBB CONOPT-PF IPOPT-CNFE
Page 24 and 25: of the feed. The reflux ratio is al
Page 26 and 27: 7 ConclusionsWith the development a
Page 28: [18] Mittelmann, H.D. and A. Pruess

MPEC Problem Formulations in Chemical Engineering Applications

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