MPEC Problem Formulations in Chemical Engineering Applications

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4.1 Automatic Reformulation of ComplementarityMPCC models written in the GAMS modeling language may be automaticallyreformulated as an NLP using several of the previously described complementaritysolution techniques. This can be accomplished with the use of the NLPECmeta-solver [8]. When called, NLPEC reformulates the complementarity constraintsof a MPCC model with a user specified reformulation, including (17)-(20). NLPEC will then call a user-specified NLP solver to solve the reformulatedmodel. The final results from the NLP solver are then translated back into theoriginal MPCC model. NLPEC verifies that the complementarities are satisfiedin the resulting solution. This is particularly important when using the PF(ρ)formulation described above, as it may not converge to a stationary point of theMPCC if the penalty parameter is too small.4.2 Comparison of Solution Strategies on MPECLibMPECLib is a set of MPEC test problems maintained by Dirkse [19]. The testset currently consists of 92 problems, which include both small scale modelsfrom the literature and several large industrial models.The performance and robustness of several NLP reformulations includingPF(ρ) with ρ = 10, Reg(ǫ) with ǫ = 10 −8 , and NCP formulation x T y = 0,x, y ≥ 0 were compared on the MPECLib test set. CONOPT (version 3.14) andIPOPT (version 3.2.3) were used to the solve the resulting NLP problems. Allreformulations were performed automatically by the NLPEC solver describedearlier. Also included in the comparison is the IPOPT-C solver, which is capableof processing the complementarity constraints without reformulation. TheCONOPT and IPOPT runs were performed entirely within the GAMS environment.However, IPOPT-C is not currently linked to GAMS. In order toperform the tests with IPOPT-C, the GAMS models were instead convertedto the AMPL modeling language format. The problems were then run usingAMPL and IPOPT-C.The results are presented in a Dolan-Morè plot. All results were obtained ona Intel Pentium 4, 1.8 GHz computer with 992 MB of RAM. This plot portraysboth robustness and relative performance of a set of solvers on a given problemset. Here we define a performance ratio calculated as:η p,s =t p,smin {t p,s : 1 ≤ s ≤ n s }(23)where n s is the number of solvers, which are run over the n p problems in set P,and where t p,s is the solution time for solver s on problem p. For all solvers sthat fail to solve problem p, we set the performance ratio to an arbitrarily largenumber. The profiles are then defined by:p s (τ) = 1 n psize {p ∈ P : η p,s ≤ τ} (24)10
and p s (τ) is the percentage of problems solved within τ ≥ η p,s times the bestperformance. This analysis can be performed automatically using the PAVERserver [18].Figure 1 shows the performance plot of different reformulations and solverstested on all 92 problems of the test set. Since the problems are inherentlynonconvex, multiple local solutions can be expected for each problem. Becauseof this, not all of the different reformulation and solver combinations convergedto the same local solutions. This makes a direct comparison of performancedifficult as the solvers often converge to different solutions. On the other hand,this figure is useful to demonstrate the robustness of the methods, as seen bylim τ →∞ p s (τ). For all but one solution method, the reformulation and solvercombination was able to solve at least 84% of all of the test problems. Themost reliable solvers were IPOPT-Reg(ǫ) (99%), CONOPT-Penalty (98%) andIPOPT-Penalty (94 %). In contrast, IPOPT with the NCP formulation (listedas IPOPT-Mult on the figure) was only able to solve 57% of all of the problems.This is not entirely unexpected as IPOPT is an interior point algorithm and theNCP formulation provides no strict interior to the NLP problem.Figure 2 shows the same reformulations and solvers tested as before excepton only 22 of the 92 test problems. These 22 problems all converged to the samesolution for all reformulations and solvers when they converged. This allows fora fair comparison of the solvers’ performance. Table 1 displays these 22 testproblems and their respective objective function values.A few observations can be made on these results. CONOPT-Reg(ǫ) performedthe fastest on 73 % of the problems but it turns out to solve only 76%of them. Instead, CONOPT-Penalty and IPOPT-Penalty (PF(ρ) are the nextbest in performance and they also prove to the most robust, solving over 90%of the problems. These turn out to be the best all-around methods.For the results in Figure 2, the CONOPT formulations take advantage ofthe active set strategy and particularly the detection and removal of degenerateconstraints. As a result, the CONOPT-Mult (with the NCP reformulation)performs well; this method is fastest on 50% of the problems but no morerobust than CONOPT-Reg. Because the test problems are not large, CONOPTprovides the best performance on this test set.On the other hand, the IPOPT formulations (IPOPT-Penalty, IPOPT-Reg,IPOPT-C) follow similar trends within Figure 2, with the penalty formulation asthe most robust. This can be explained by the similarities among these methods,as analyzed in [15]. Finally, IPOPT-Mult (with the NCP reformulation) is theworst performer as it cannot remove dependent constraints and therefore suffersmost from the inherent degeneracies in MPCCs.5 Process Engineering Examples of MPCCsComplementarity applications in chemical engineering are widespread for modelingdiscrete decisions. Their use for modeling and optimizing split range controllersand tiered splitters has been known for some time [7]. However, with a11
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Page 8 and 9: RegEq(ǫ): min f(w) (19a)s.t. h(w)
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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
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Page 28: [18] Mittelmann, H.D. and A. Pruess

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