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.
|Title of host publication
|Integer Programming and Combinatorial Optimization - 22nd International Conference, IPCO 2021, Proceedings
|Mohit Singh, David P. Williamson
|Springer Science and Business Media Deutschland GmbH
|Number of pages
|Published - May 5 2021
|22nd International Conference on Integer Programming and Combinatorial Optimization, IPCO 2021 - Virtual, Online
Duration: May 19 2021 → May 21 2021
|Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
|22nd International Conference on Integer Programming and Combinatorial Optimization, IPCO 2021
|5/19/21 → 5/21/21
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- Bipartite bilinear sets