We investigate the generalized red refinement for n-dimensional simplices that dates back to Freudenthal (Ann Math 43(3):580–582, 1942) in a mixed-integer nonlinear program (\({\textsc {MINLP}}\)) context. We show that the red refinement meets sufficient convergence conditions for a known \({\textsc {MINLP}}\) solution framework that is essentially based on solving piecewise linear relaxations. In addition, we prove that applying this refinement procedure results in piecewise linear relaxations that can be modeled by the well-known incremental method established by Markowitz and Manne (Econometrica 25(1):84–110, 1957). Finally, numerical results from the field of alternating current optimal power flow demonstrate the applicability of the red refinement in such \({\textsc {MIP}}\)-based \({\textsc {MINLP}}\) solution frameworks.

我们在混合整数非线性程序 ( \({\textsc {MINLP}}\) ) 中研究了可追溯到 Freudenthal (Ann Math 43(3):580–582, 1942) 的n维单纯形的广义红色细化语境。我们表明红色细化满足已知\({\textsc {MINLP}}\) 的充分收敛条件 解决方案框架基本上基于解决分段线性松弛。此外，我们证明应用此细化程序会导致分段线性松弛，可以通过 Markowitz 和 Manne 建立的著名增量方法（Econometrica 25(1):84–110, 1957）建模。最后，交流电优化潮流领域的数值结果证明了红色细化在基于\({\textsc {MIP}}\)的\({\textsc {MINLP}}\) 解决方案框架中的适用性。