Discretization-based methods have been proposed for solving nonconvex optimization problems with bilinear terms. These methods convert the original nonconvex optimization problems into mixed-integer linear programs (MILPs). Compared to a wide range of studies related to methods to convert nonconvex optimization problems into MILPs, research on tightening the resulting MILP models is limited. In this paper, we present tightening constraints for the discretization-based MILP models for the pooling problem. Specifically, we study tightening constraints derived from upper bounds on bilinear term and exploiting the structures resulting from the discretization. We demonstrate the effectiveness of our constraints, showing computational results for MILP models derived from different formulations for (1) the pooling problem and (2) discretization-based pooling models. Computational results show that our methods reduce the computational time for MILP models on CPLEX 12.10. Finally, we note that while our methods are presented in the context of the pooling problem, they can be extended to address other nonconvex optimization problems with upper bounds on bilinear terms.
翻译:以双线性术语提出以分解法为基础的方法来解决非混凝土优化问题。这些方法将原非混凝土优化问题转换成混合整流线性程序(MILPs)。与将非混凝土优化问题转换成MILP方法的广泛研究相比,关于收紧由此产生的MILP模型的研究有限。在本文中,我们对基于离散的MILP模型的集合问题提出了更严格的限制。具体地说,我们研究从双线性术语上限产生的限制,并利用离散结构产生的限制。我们展示了我们的限制的有效性,展示了从(1)联合问题和(2)基于离散的集合模型的不同配方得出的MILP模型的计算结果。比较结果显示,我们的方法减少了以CPLEX 12.10为主的MILP模型的计算时间。最后,我们注意到,虽然我们的方法是在集合问题的背景下提出,但可以扩大到解决具有双线性术语上框的其他非凝固化优化问题。