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Lifting Convex Inequalities for Bipartite Bilinear Programs

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Book cover Integer Programming and Combinatorial Optimization (IPCO 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12707))

Abstract

The goal of this paper is to derive new classes of valid convex inequalities for quadratically constrained quadratic programs (QCQPs) through the technique of lifting. Our first main result shows that, for sets described by one bipartite bilinear constraint together with bounds, it is always possible to lift a seed inequality that is valid for a restriction obtained by fixing variables to their bounds, when the lifting is accomplished using affine functions of the fixed variables. In this setting, sequential lifting involves solving a non-convex nonlinear optimization problem each time a variable is lifted, just as in Mixed Integer Linear Programming. To reduce the computational burden associated with this procedure, we develop a framework based on subadditive approximations of lifting functions that permits sequence independent lifting of seed inequalities for separable bipartite bilinear sets. In particular, this framework permits the derivation of closed-form valid inequalities. We then study a separable bipartite bilinear set where the coefficients form a minimal cover with respect to right-hand-side. For this set, we derive a “bilinear cover inequality”, which is second-order cone representable. We argue that this bilinear covering inequality is strong by showing that it yields a constant-factor approximation of the convex hull of the original set. We study its lifting function and construct a two-slope subadditive upper bound. Using this subadditive approximation, we lift fixed variable pairs in closed-form, thus deriving a “lifted bilinear cover inequality” that is valid for general separable bipartite bilinear sets with box constraints.

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Notes

  1. 1.

    We use the term bipartite, perhaps redundantly, to highlight that variables can be divided into two groups, such that any degree two term comes from product of variables one each from these two groups [22].

  2. 2.

    We say “almost”, since there are non-packing examples, such as \(S:= \{x,y \in [0, 1]^2\,|\, x_1y_1 - 100x_2y_2 \ge -98\}\), where there is no partition that yields a minimal cover. Such sets are “overwhelmingly” like a packing set; in the case of the example, it is a perturbation of the packing set \(\{x_2, y_2 \in [0, 1]\,|\, 100x_2y_2\le 98\}\). For such sets it is not difficult to show that \(conv (S)\) is polyhedral.

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Authors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Xiaoyi Gu & Santanu S. Dey

  2. University of Minnesota, Minneapolis, MN, 55455, USA

    Jean-Philippe P. Richard

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  1. Xiaoyi Gu
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  2. Santanu S. Dey
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  1. Georgia Institute of Technology, Atlanta, GA, USA

    Mohit Singh

  2. Cornell University, Ithaca, NY, USA

    David P. Williamson

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Gu, X., Dey, S.S., Richard, JP.P. (2021). Lifting Convex Inequalities for Bipartite Bilinear Programs. In: Singh, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2021. Lecture Notes in Computer Science(), vol 12707. Springer, Cham. https://doi.org/10.1007/978-3-030-73879-2_11

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