(253e) On Piecewise Under- and over-Estimators of Fractional Terms
We consider various ways of constructing piecewise under- and over-estimators of fractional terms. These estimators are obtained using outer-linearization of bilinear terms [1 â 4], quadratic/linear underestimator [1,3], and outer-approximation followed by outer-linearization, an approach recently proposed in . Since these relaxations rely heavily on bounds, we introduce binary variables, and use the incremental cost formulation  along with the reformulation-linearization technique  to develop piece-wise relaxations. This step differs from the standard approach of constructing IP relaxations via piecewise relaxations of bilinear terms obtained after cross-multiplying the fractional term. We show that using the proposed IP relaxations we are able to solve, for the first time, to near optimality, an MINLP that identifies optimal multicomponent distillation sequences. The fractional terms in this formulation arise from Underwood equations and the combinatorial choices model the structural specification of the distillation configuration. We present computational results, in the context of this application, comparing the efficacy of various estimators for the fractional terms.
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