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Quadratic reformulations for the optimization of pseudo-boolean functions

Abstract

We investigate various solution approaches for the uncon- strained minimization of a pseudo-boolean function. More precisely, we assume that the original function is expressed as a real-valued polynomial in 0-1 variables, of degree three or more, and we consider a generic family of two-step ap- proaches for its minimization. First, a quadratic reformu- lation step aims at transforming the minimization problem into an equivalent constrained or unconstrained quadratic 0-1 minimization problem (where “equivalent” means here that a minimizer of the original function can be easily de- duced from a minimizer of the reformulation). Second, an optimization step handles the obtained equivalent quadratic problem. We provide a unified presentation of several quadratic re- formulation schemes proposed in the literature, e.g., (An- thony et al. 2017; Buchheim and Rinaldi 2007; Rodr ́ıguez- Heck 2018; Rosenberg 1975), and we review several meth- ods that can be applied in the optimization step, including a standard linearization procedure (Fortet 1959) and more elaborate convex quadratic reformulations, as in (Billionnet and Elloumi 2007; Billionnet, Elloumi, and Lambert 2012, 2016; Elloumi, Lambert, and Lazare 2021). We discuss the impact of the reformulation scheme on the efficiency of the optimization step and we illustrate our discussion with some computational results on different classes of instances.
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Dates and versions

hal-03537270 , version 1 (20-01-2022)

Identifiers

  • HAL Id : hal-03537270 , version 1

Cite

Yves Crama, Sourour Elloumi, Amélie Lambert, Elisabeth Rodriguez-Heck. Quadratic reformulations for the optimization of pseudo-boolean functions. International Symposium on Artificial Intelligence and Mathematics ISAIM 2022, Jan 2022, Fort Lauderdale, United States. ⟨hal-03537270⟩
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