Mathematikgebäude, Raum M 511

Partial outer convexication has been introduced as a relaxation technique for MINLPs that are constrained by ordinary dierential 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  > 0. Advantageously, it does so in linear time w.r.t the number of cells that make up the rounding grid. Rening the rounding grid induces an improved approximation of the relaxed control problem's trajectory in a suitable weak topology. If the dierential equation exhibits sucient 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 dierentiability 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 renements to employ a similar chain of arguments for a class of elliptic PDE systems. We give a sucient condition for such desirable ordering strategies and show that they are satised by the approximants of space-lling curves.

M.Sc. Paul Manns

TU Braunschweig