A new Lagrangean approach to the pooling problem

APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes

“…Nonconvex bilinear terms in the standard pooling problem arise from tracking the levels of linearlyblending fuel qualities about the pooling nodes to meet constraints on the composition of the final products [Visweswaran, 2009]. Among the many notable contributions towards solving the standard pooling problem [Adhya et al, 1999, Almutairi and Elhedhli, 2009, Audet et al, 2004, Ben-Tal et al, 1994, Chakraborty, 2009, Quesada and Grossmann, 1995, Foulds et al, 1992, Greenberg, 1995, Haverly, 1978, Lasdon et al, 1979, Lodwick, 1992, Pham et al, 2009, Tawarmalani and Sahinidis, 2002, the most directly relevant to the work presented in this paper and the computational tool APOGEE are those of: Floudas and Visweswaran , who were the first to rigorously solve the pooling problem to global optimality; Foulds et al [1992], who developed a linear relaxation of the QCQP by replacing each bilinear term with their convex and concave hulls [Al-Khayyal andFalk, 1983, McCormick, 1976]; Ben-Tal et al [1994], who introduced an alternative q-formulation of the pooling problem that often has fewer nonconvex bilinear terms than the original p-formulation [Audet et al, 2004]; and Tawarmalani and Sahinidis [2002], who showed that augmenting the q-formulation with reformulation-linearization technique cuts Adams, 1999, Sherali andAlameddine, 1992] proposed by Quesada and Grossmann [1995] produces a linear relaxation of the pooling problem that strictly dominates both the p-and q-formulations. Depending on the formulation, the standard pooling problem can be classified as a linear objective with quadratic constraints (p-formulation) or a quadratic objective with quadratic constraints (q-and pq-formulations).…”

GloMIQO: Global mixed-integer quadratic optimizer

“…, M} Major applications of MIQCQP include quality blending in process networks, separating objects in computational geometry, and portfolio optimization in finance. Specific instantiations of MIQCQP in process networks optimization problems include: pooling problems [5,12,20,28,53,65,66,67,79,93,97,98,100,101,107,137,139], distillation sequences [8,54,58], wastewater treatment and total water systems [9,13,22,29,50,62,71,73,108,109], hybrid energy systems [23,24,49], heat exchanger networks [39,56], reactorseparator-recycle systems [75,76], separation systems [119], data reconciliation [115], batch processes [86], crude oil scheduling [78,80,81,104,103], and natural gas production [82,…”

New multi-commodity flow formulations for the pooling problem

“…Constraint (11), on the other hand, ensures that at every pool, the total incoming flow originating from an input must equal the total outgoing flow originating from the same input. In fact, using (10) and (13), we can show that (11) implies (1):…”

“…(11) can be viewed as a disaggregated flow conservation constraint. Note that (1) ensures that at every pool, the total incoming flow must equal the total outgoing flow, regardless of the origin of the flows.…”