Abstract
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.
Highlights
Solving general mixed-integer nonlinear program (MINLP) is to this day a very challenging task
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
We showed that the generalized red refinement for n-dimensional simplices can be utilized for solving MINLP problems by piecewise linear relaxations
Summary
Solving general MINLPs is to this day a very challenging task. The backbone of most approaches in this context is still branch-and-bound. They show that the classical longest-edge bisection fulfills these conditions and is suitable for such a solution framework They prove that triangulations that are constructed by successively applying the longest-edge bisection lead to piecewise linear relaxations that can always be modeled by the already mentioned generalization of the incremental method. We extend this result by another refinement strategy for n-dimensional simplices: the generalized red refinement introduced by Freudenthal in [5].
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