Primal-Dual Symmetric Scale-Invariant Square-Root Fields for Isotropic Self-Scaled Barrier Functionals

Raphael Hauser

Square-root fields are differentiable operator fields used in the construction of target direction fields for self-scaled conic programming, a unifying framework for primal-dual interior-point methods for linear programming, semidefinite programming and second-order cone programming. In this article we investigate square-root fields for so-called isotropic self-scaled barrier functionals, i.e. self-scaled barrier functionals that are invariant under rotations of their conic domain of definition. We prove a structure theorem for so-called congruent square-root fields for isotropic self-scaled barrier functionals in terms of the irreducible decomposition of their domain of definition. Using this structure theorem, we then investigate primal-dual symmetry and scale-invariance of such square-root fields. In our main theorem we show that these two assumptions together with one additional natural invariance property (so-called canonical reduction) dramatically reduce the degree of freedom in the choice of square-root fields that satisfy these properties, but that such square-root fields always exist.

Numerical Analysis Report DAMTP 1999/NA16, Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, England CB3 9EW.