### Optimization and Systems Theory Seminar

Friday, August 24 2007, 11.00-12.00, Room 3721, Lindstedtsvägen 25

** Mathias Stolpe**

** Department of Mathematics, Technical University of Denmark (DTU)**

E-mail: m.stolpe@mat.dtu.dk

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GLOBAL OPTIMIZATION OF DISCRETE TOPOLOGY DESIGN PROBLEMS

A classical problem within the field of structural topology optimization is
to find the stiffest structure subject to multiple loads and a bound on the
volume (or weight) of the structure. We minimize a weighted average of the
compliances, i.e. the inverse of the stiffness. The design variables describe
the cross sectional areas of the bars in a truss or fiber directions in a
structure made of laminated composites. This class of problems is well-studied
for continuous variables. We consider here the situation that the variables are
discrete.

Our goal is to compute guaranteed globally optimal structures. We present a
method for the computation of a global optimizer of the underlying non-convex
discrete problem. The method is a finitely convergent nonlinear branch and cut
method tailored to solve large-scale instances of the original discrete problem.
The branch and cut algorithm is based on solving a sequence of continuous
relaxations, which are obtained by relaxing the discreteness requirements. The
main effect of this approach lies in the fact that these relaxed problems can be
equivalently reformulated as all-quadratic convex problems and thus can be
efficiently solved to global optimality.

The presented nonlinear branch and cut method is numerically compared to a
commercial branch and cut method applied to a convex mixed 0-1 equivalent
reformulation of the original discrete problem. The commercial software solves
significantly more relaxations. The main reason for this behavior is explained
by comparing the strength of the relaxations used. We present global optimal
solutions to several large-scale numerical examples.

Calendar of seminars
*Last update: August 14, 2007 by
Marie Lundin.
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