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Technische Universität Dortmund
Fak. Mathematik, LS X
44221 Dortmund

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Technische Universität Dortmund
Fak. Mathematik, LS X
Vogelpothsweg 87
44227 Dortmund


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Vortrag am 29.11.2018 im Rahmen des Oberseminars Numerische Analysis und Optimierung

Vortragender: M.Sc. Paul Manns, TU Braunschweig

Zeit: 14.15 Uhr

Ort: Mathematikgebäude, Raum M 511

Titel: Approximation properties of Sum-Up Rounding

Abstract: Partial outer convexi fication has been introduced as a relaxation technique for MINLPs that are constrained by ordinary diff erential equations. The family of Sum-Up Rounding algorithms allows to approximate feasible points of the continuously-valued relaxation with discrete ones that are feasible up to an arbitrarily small delta  > 0. Advantageously, it does so in linear time w.r.t the number of cells that make up the rounding grid. Refi ning the rounding grid induces an improved approximation of the relaxed control problem's trajectory in a suitable weak topology. If the diff erential equation exhibits sufficient regularity, the corresponding sequence of state vectors can be shown to converge in norm. We are able to prove the approximation property for ODEs and for time-dependent semilinear PDEs under mild regularity assumptions on the solution trajectory of the PDE. In particular, previous requirements of di fferentiability and uniformly bounded derivatives on the involved functions can be dropped. Regarding PDE-Constrained MINLPs with integer variables distributed in more than one dimension, we can combine an appropriate grid re nement and a feasible ordering strategy of the grid cells during the refi nements to employ a similar chain of arguments for a class of elliptic PDE systems. We give a sufficient condition for such desirable ordering strategies and show that they are satis fied by the approximants of space- filling curves.