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