Convex Hull Discretization Approach to the Global Optimization of Pooling Problems

Abstract

The pooling problem is an important optimization problem that is encountered in process operation and scheduling. Because of the presence of bilinear terms, the traditional formulation is nonconvex. Consequently, there is a need to develop computationally efficient and easy-to-implement global-optimization techniques. In this paper, a new approach is proposed based on three concepts: linearization by discretizing nonlinear variables, preprocessing using implicit enumeration of the discretization to form a convex-hull which limits the size of the search space, and application of integer cuts to ensure compatibility between the original problem and the discretized formulation. The continuous quality variables contributing to the bilinear terms are first discretized. The discretized problem is a mixed integer linear program (MILP) which is globally solvable in a computationally effective manner using the branch and bound method. The merits of the proposed approach are illustrated with case studies from literature and comparison with published results.

GloMIQO: Global mixed-integer quadratic optimizer

“…The GloMIQO reformulation uses the observation that disaggregating bilinear terms tightens the relaxation of MIQCQP and actively takes advantage of any redundant linear constraints added to the model. It is standard to use termwise convex/concave envelopes [11,91] to relax MIQCQP, but many tighter relaxations have been developed based on: polyhedral facets of edge-concave multivariable term aggregations [17,26,34,94,95,96,99,111,130,131,132], eigenvector projections [38,106,113,122], piecewise-linear underestimators [29,65,66,73,93,98,99,100,101,107,119,139], outer approximation of convex terms [32,48,47], and semidefinite programming (SDP) relaxations [16,25,35,122,121]. GloMIQO incorporates several of these advanced relaxations.…”

“…Our previous work integrated piecewise-linear underestimators into the global optimization branch-and-bound tree based on computational experimentation determining the best formulations for these underestimators [29,65,66,73,93,98,99,100,101,107,119,139]. After extensive testing on process networks and point packing problems [99], we suggested that using the piecewise-linear relaxation scheme increases the probability that MIQCQP will solve to ε-global optimality within a given time limit.…”

“…, 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,…”

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).…”

“…Hasan and Karimi [2010] studied the possibility of bivariate partitioning, that is, segmenting both variables participating in each bilinear term. Other groups who have used piecewise-linear underestimators include: Bergamini et al [2008] in their Outer Approximation for Global Optimization Algorithm; Saif et al [2008], in a reverse osmosis network case study; and Pham et al [2009], in a fast-solving algorithm that generates near-optimal solutions.…”