# Product Disaggregation in Global Optimization and Relaxations of Rational Programs

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## Abstract

We consider the product of a single continuous variable and the sum of a number of continuous variables. We show that “product disaggregation” (distributing the product over the sum) leads to tighter linear programming relaxations, much like variable disaggregation does in mixed-integer linear programming. We also derive closed-form expressions characterizing the exact region over which these relaxations improve when the bounds of participating variables are reduced.

In a concrete application of product disaggregation, we develop and analyze linear programming relaxations of rational programs. In the process of doing so, we prove that the task of bounding general linear fractional functions of 0–1 variables is \(\mathcal{N}\mathcal{P}\)-hard. Finally, we present computational experience to demonstrate that product disaggregation is a useful reformulation technique for global optimization problems.

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