SIAM Journal on Optimization, Volume 30, Issue 2, Page 1582-1609, January 2020.
We investigate new convex relaxations for the pooling problem, a classic nonconvex production planning problem in which input materials are mixed in intermediate pools, with the outputs of these pools further mixed to make output products meeting given attribute percentage requirements. Our relaxations are derived by considering a set which arises from the formulation by considering a single product, a single attibute, and a single pool. The convex hull of the resulting nonconvex set is not polyhedral. We derive valid linear and convex nonlinear inequalities for the convex hull and demonstrate that different subsets of these inequalities define the convex hull of the nonconvex set in three cases determined by the parameters of the set. In a preliminary computational study we find that the inequalities can significantly strengthen the convex relaxation of the well-known $pq$-formulation of the pooling problem on one class of test instances, but have limited effect on another class.

中文翻译：

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合并问题的强凸非线性松弛

SIAM优化杂志，第30卷，第2期，第1582-1609页，2020年1月。
我们研究了合并问题的新凸松弛，这是一个经典的非凸生产计划问题，在该问题中，将输入物料混合在中间池中，并且进一步混合这些池的输出以使输出产品满足给定的属性百分比要求。我们的放松是通过考虑一组配方而产生的，而这些配方是通过考虑单个产品，单个礼品和单个池而得出的。所得非凸集的凸包不是多面体。我们得出凸包的有效线性和凸非线性不等式，并证明在三种情况下，这些不等式的不同子集定义了非凸集的凸包。