Self-scaled barriers for semidefinite programming

25 May 2000
Dr Raphael Hauser
I am going to show that all self-scaled barriers for the cone of symmetric positive semidefinite matrices are of the form $X\mapsto -c_1\ln\det X +c_0$ for some constants $c_1$ > $0,c_0 \in$ \RN. Equivalently one could state say that all such functions may be obtained via a homothetic transformation of the universal barrier functional for this cone. The result shows that there is a certain degree of redundancy in the axiomatic theory of self-scaled barriers, and hence that certain aspects of this theory can be simplified. All relevant concepts will be defined. In particular I am going to give a short introduction to the notion of self-concordance and the intuitive ideas that motivate its definition.
  • Computational Mathematics and Applications Seminar