Piecewise relaxations are powerful techniques to compute strong dual bounds for (mixed-integer) quadratically constrained problems and provide starting points that can be used by local nonlinear solvers to generate high-quality primal solutions. They work by first partitioning the domain of one of the variables in every quadratic or bilinear term and then selecting the optimal interval in the partition through binary variables. A variety of formulations have been proposed that can be distinguished based on how the number of binary variables scales with the number of intervals. In this paper, we compare linear and logarithmic partitioning schemes that can be derived from the piecewise McCormick relaxation (PCM) or the multiparametric disaggregation technique (MDT). Specifically, we propose MDT for a generic numeric representation system, showing that it becomes a linear partitioning scheme when considering a single digit and varying the basis, and a logarithmic partitioning scheme when fixing the basis and varying the number of digits. Through the solution of 25 benchmark instances for the pooling problem, considering relaxations for the p-, q- and tp-formulations, we show that it is better to rely on the linear scheme from PCM for a small number of intervals and on the logarithmic scheme from base-2 MDT when the number is large. The results also show that significantly stronger dual bounds can be obtained compared to commercial global optimization solvers BARON and GloMIQO.

分段松弛是一种强大的技术，可以计算（混合整数）二次约束问题的强对偶边界，并提供可以被局部非线性求解器用来生成高质量原始解的起点。它们的工作方式是先对每个二次项或双线性项中的一个变量的域进行分区，然后通过二进制变量在分区中选择最佳区间。已经提出了可以根据二元变量的数量如何随着间隔的数量缩放来区分的各种公式。在本文中，我们比较了可以从分段McCormick弛豫（PCM）或多参数分解技术（MDT）派生的线性和对数划分方案。具体来说，我们为通用数字表示系统提出了MDT，表示当考虑一位数字并改变基数时，它成为线性分区方案；而当确定基数并改变数字数时，它成为对数分区方案。通过针对池问题的25个基准实例的解决方案，考虑对p，q和tp公式的弛豫，我们表明最好是在较小的时间间隔和对数上依靠PCM的线性方案数量大时，从base-2 MDT的方案。结果还表明，与商业全局优化求解器BARON和GloMIQO相比，可以获得明显更强的对偶边界。通过针对池问题的25个基准实例的解决方案，考虑对p，q和tp公式的弛豫，我们表明最好是在较小的时间间隔和对数上依靠PCM的线性方案数量大时，从base-2 MDT的方案。结果还表明，与商业全局优化求解器BARON和GloMIQO相比，可以获得明显更强的对偶边界。通过针对池问题的25个基准实例的解决方案，考虑对p，q和tp公式的弛豫，我们表明最好是在较小的时间间隔和对数上依靠PCM的线性方案数量大时，从base-2 MDT的方案。结果还表明，与商业全局优化求解器BARON和GloMIQO相比，可以获得明显更强的对偶边界。