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362673 Tightening Piecewise Mccormick Relaxations through Partition-Dependent Bounds for Non-Partitioned Variables

The simplest and most common type of nonlinear constraint in Chemical Engineering arises when mixing multicomponent streams of different properties. Blending constraints appear in crude oil operations in refineries (Lee et al. 1996, Jia et al. 2003), in the blending of liquid fuels (Jia and Ierapetritou 2003, Kolodziej et al. 2013b), in the design of distributed wastewater treatment systems (Galan and Grossmann 1998, Meyer and Floudas 2006, Teles et al. 2012) and integrated water networks (Karuppiah and Grossmann 2006, Faria and Bagajewicz 2012, Rubio-Castro et al. 2013). Bilinear terms also arise in the trim loss problem in paper plants (Harjunkoski et al. 1998, Zorn and Sahinidis 2013). Bilinear terms are nonconvex and therefore give rise to a variety of local solutions that may be far from the global optimum.

In order to find rigorous global optimal solutions to bilinear problems, alternative algorithms can be used that have in common the generation of linear or mixed-integer linear relaxations of the original problem. A well-known example of an MILP relaxation is the piecewise McCormick envelopes of Grossmann and co-workers (Bergamini et al. 2005, Karuppiah and Grossmann 2006). While piecewise McCormick relaxation can prove global optimality for a typically large number of partitions, the resulting MILP problem quickly becomes intractable due to the linear growth of binary variables with the number of partitions (Kolodziej et al. 2013a). A tighter formulation using the same number of binary variables is thus needed.

Piecewise McCormick relaxation works by diving the domain of one of the variables in each bilinear term into a given number of partitions. A tighter relaxation, compared to the standard linear McCormick (1976) envelopes, results from considering partition-dependent lower and upper bounds for the partitioned variables. We now propose to use partition-dependent bounds also for the non-partitioned variables so as to further improve the quality of the relaxation. These can be determined through optimality-based bound contraction (see for instance Castro and Grossmann 2014), which involves solving two linear problems per partition and per variable. While hundreds or even thousands of linear bound contracting problems may be involved, the benefit from a tighter formulation more than compensates the additional computational time. Once the solution of the tightened relaxation is found, it is used as a starting point for solving the original nonlinear problem with a fast local solver, so as to find a near optimal solution and to compute a rigorous optimality gap.

The performance of the new algorithm is illustrated with a set of small test problems and larger water-using network design problems from the literature. For the latter, the results show that the proposed algorithm outperforms state-of-the-art commercial solvers BARON (Tawarmalani and Sahinidis 2005) and GloMIQO (Misener and Floudas 2013). This is an indication that piecewise relaxation schemes and optimality-based bound contraction should be used to a greater extent.

**Acknowledgments:** Financial support from the Luso-American Foundation under the 2013 Portugal-U.S. Research Networks program and from Fundação para a Ciência e Tecnologia through the Investigador FCT 2013 program.

**References:**

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