On refinement strategies for solving MINLPs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsc {MINLP}\mathrm{s}}$$\end{document}  by piecewise linear relaxations: a generalized red refinement

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 (MINLP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsc {MINLP}}$$\end{document}) context. We show that the red refinement meets sufficient convergence conditions for a known MINLP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsc {MINLP}}$$\end{document} 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 MIP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsc {MIP}}$$\end{document}-based MINLP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsc {MINLP}}$$\end{document} solution frameworks.