262128 New Global Optimization Algorithm for Water-Using Networks
Design approaches for water networks can be classified into pinch analysis and mathematical optimization techniques. Pinch technology provides a comprehensive understanding of the problem at hand through simple graphical methods but is difficult to apply to large problems featuring multiple contaminants or to optimization goals besides flowrate minimization. On the other hand, rigorous optimization approaches through mathematical programming models start with the generation of a complex superstructure involving all design possibilities. A major advantage is their wider scope, which is why this work focuses on mathematical models.
Water networks (Bagajewicz, 2000; Jezowski, 2010) are a special type of process network problems (Quesada & Grossmann, 1995) that can be formulated as bilinear programs. Another example is the pooling problem (Haverly, 1978; Meyer & Floudas, 2006; Misener & Floudas, 2011). The non-convex bilinear terms arise in the mixing of the streams with different properties and are known to give rise to a multiplicity of local optima, which prevent gradient based solvers from certifying optimality of the nonlinear program (NLP). Further complexity will arise from binary decisions associated to alternative process units and/or connecting pipelines, leading to a mixed-integer nonlinear program (MINLP).
The most common global optimization algorithms are based on spatial branch-and-bound (Wicaksono and Karimi, 2008) and involve multiple formulations of a lower bounding problem that is a relaxation of the original bilinear problem. The relaxations are frequently based on the standard McCormick (1976) envelopes, leading to a linear problem (LP), or on piecewise envelopes (Bergamini et al. 2005; Karuppiah & Grossmann 2006). In the latter, the domain of the variables is divided ab initio into a given number of partitions, with the purpose of generating multiple McCormick envelopes that will provide a tighter relaxation. The optimal set of partitions is identified through binary variables leading to a mixed-integer linear programming problem (MILP).
We have recently proposed an alternative global optimization approach called multiparametric disaggregation (Teles et al. 2011) and applied it to the water-using networks (WUN) design problem (Teles et al. 2012). Through the discretization of the domain of one of the variables of the bilinear term, the original problem can be approximated to any desired accuracy. The discretization is based on the concept of significant digits and can be performed with respect to different numeric representation systems (e.g. decimal, binary). An important aspect is that the number of binary variables scales linearly, as opposed to exponentially (piecewise McCormick), with increasing accuracy.
In this paper, the multiparametric disaggregation (MDT) approach is upgraded by proposing a rigorous relaxation to the bilinear problem arising in the optimal design of WUNs, enabling us to generate a lower rather than an upper bound. The ability of becoming increasingly tight with the increase in accuracy is kept, providing the natural iterations of a novel global optimization algorithm. Through the solution of several test cases from the literature, we show that the new algorithm terminates with lower optimality gaps than the ones determined by the commercial solver BARON in 89% of the cases. Since MDT works by discretizing half of the variables appearing in bilinear terms, we test the two alternatives of parameterizing the concentrations and flowrates, with the results favoring the former.
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