FWF Project P22239 - Symmetries in global optimization

by Hermann Schichl

This project aims at the development of a theoretical framework accompanied by algorithms implemented as modules for the public domain global optimization package ``COCONUT Environment'' for handling discrete and continuous symmetries in global optimization problems. Therefore, a generalized interval analysis on Lie groups and their representations will be developed. This will be used to solve symmetric global optimization problems on the lower dimensional orbifold. The newly researched algorithms in the public domain global optimization package ``COCONUT Environment''.

We plan to develop a generalization of interval analysis to Lie algebras and Lie groups and their linear representations in order to do rigorous Lie group numerics including explicit error analysis. Mainly, we will focus our attention to matrix Lie groups and their action on submanifolds of $\Rz^{n}$. Then, we intend to apply this rigorous methods for reducing symmetric optimization problems to the orbifold, which is generated by factoring out the Lie group action of the feasible set. We will calculate proper optimality conditions for the problem on the orbifold, hereby eliminating the degeneracy caused by the continuous symmetry. Using these optimality conditions and the rigorous computation techniques, we intend to compute inclusion and exclusion regions generalizing those already known. We will research adapted splitting strategies for the branch and bound scheme, and, if necessary, adaptions needed to local optimization (like search path generation on the orbifold) for increasing the performance of the local optimizer. Implications to constraint propagation will also be considered.

Hermann Schichl (Hermann.Schichl@univie.ac.at)