(315f) Tightening Mccormick Relaxations Via Reformulation of Intermediate Functions into Schema | AIChE

(315f) Tightening Mccormick Relaxations Via Reformulation of Intermediate Functions into Schema


Wilhelm, M. - Presenter, University of Connecticut
Stuber, M., University of Connecticut
Ernst, R., University of Connecticut
Generalized McCormick relaxations [2] can present significant boons in global optimization applications. This framework allows for the relaxation and subsequent optimization of algorithmically-defined functions that naturally arise during flowsheet modeling. Relaxations of functions considered under this theory exhibit quadratic convergence, helping to reduce the degree of clustering in spatial branch-and-bound algorithms [1-4]. Moreover, this simulation-based feasible-path approach enables the solution of flowsheeting problems in a reduced-dimension decision space, in contrast to the higher-dimensionality “lifting” of the decision space required by state-of-the-art auxiliary variable methods (AVMs) [5-6].

McCormick relaxations are typically calculated by overloading common arithmetic operators [2,3] in object-oriented programming languages [7]. Most recent research in McCormick-related relaxations have focused on providing improvements to the properties of these calculations to yield tighter relaxations, faster. Mitsos et al. developed tighter multivariate relaxations [8]. Khan et al. extended this to develop differentiable analogs [9]. Recently, Najman et al. extended this standard framework; provided tighter relaxations by using affine relaxations to shrink the associated interval bounds [10]. This represents one of the first usages of a reformulation of the directed-acyclic-graphs (DAGs) to tighten McCormick relaxations.

Analysis of the DAG representation of functions has been used in many applications ranging from constraint-satisfaction problems to scheduling [11,12]. Recently, the usage of convex-transformable functions was explored within the BARON context resulting in a significant improvement to performance on standard benchmark libraries [13]. Khajavirad et al. identified a number of regular expressions in DAGs that allow for tighter relaxations when relaxed directly [14]. In this work, we discuss the potential of tightening McCormick relaxations via the reformulation of intermediate functions.

Rather than define a series of regular expressions, we define a series of schema based on various functional attributes (convexity information). A graph theoretic approach to identifying potential intermediate function reformulations is developed that allows for reformulation presented in terms of cut vertices [15-17]. At these potential points of reformulation, a series of general tests based on interval arithmetic are applied to determine the function schema. This approach makes use of the abstract-syntax tree (AST) feature of Julia [18] to dynamically build tighter convex and concave relaxations of expressions belonging to these schemata. A series of examples from both the COCONUT library and extant literature are presented [19].

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