We investigate the generalized red refinement for n-dimensional simplices that dates back to Freudenthal in a mixed-integer nonlinear program (MINLP) context. We show that the red refinement meets sufficient convergence conditions for a known 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. Finally, numerical results from the field of alternating current optimal power flow demonstrate the applicability of the red refinement in such MIP-based MINLP solution frameworks.
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Discrete Optimization, Cauerstr. 11, 91058 Erlangen, Germany, July 2020