SIAM Journal on Optimization, Volume 30, Issue 2, Page 1339-1365, January 2020.
One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well-known McCormick relaxation for a product of two variables $x$ and $y$ over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of $xy$ can be much tighter when computed over a strict, nonrectangular subset of the box. In order to exploit this in practice, we propose computing valid linear inequalities for the projection of the feasible region onto the $x$-$y$-space by solving a sequence of linear programs akin to optimization-based bound tightening. These valid inequalities allow us to employ results from the literature to strengthen the classical McCormick relaxation. As a consequence, we obtain a stronger convexification procedure that exploits problem structure and can benefit from supplementary information obtained during the branch-and-bound algorithm such as an objective cutoff. We complement this by a new bound tightening procedure that efficiently computes the best possible bounds for $x$, $y$, and $xy$ over the available projections. Our computational evaluations using the academic solver SCIP exhibit that the proposed methods are applicable to a large portion of the public test library MINLPLib and help to improve performance significantly.

中文翻译：

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使用二维投影来增强双线性项的分离和传播

SIAM优化杂志，第30卷，第2期，第1339-1365页，2020年1月。
混合整数非线性规划求解器中最基本的成分之一是众所周知的McCormick松弛，它在框约束域上生成两个变量$ x $和$ y $的乘积。本文的出发点是这样一个事实，即在严格的非矩形子集上进行计算时，$ xy $图的凸包会更紧密。为了在实践中利用这一点，我们提出了通过解决一系列类似于基于优化的约束紧缩的线性程序来计算有效区域在$ x $-$ y $-空间上的投影的有效线性不等式。这些有效的不等式使我们能够利用文献资料来加强经典的麦考密克松弛。作为结果，我们获得了一个更强大的凸化程序，该程序利用问题结构，并且可以从分支定界算法（例如，目标截止）中获得的补充信息中受益。我们通过新的边界收紧程序对它进行补充，该程序可以有效地计算$ x $，$ y $和$ xy $在可用预测范围内的最佳边界。我们使用学术求解器SCIP进行的计算评估表明，所提出的方法适用于公共测试库MINLPLib的很大一部分，并有助于显着提高性能。