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

Square-root elds are diierentiable operator elds used in the construction of target direction elds for self-scaled conic programming, a unifying framework for primal-dual interior-point methods for linear programming, semideenite programming and second-order cone programming. In this article we investigate square-root elds for so-called isotropic self-scaled barrier functionals, i.e. self-scaled barrier functionals that are invariant under rotations of their conic domain of deenition. We prove a structure theorem for so-called congruent square-root elds for isotropic self-scaled barrier functionals in terms of the irreducible decomposition of their domain of deenition. Using this structure theorem, we then investigate primal-dual symmetry and scale-invariance of such square-root elds. 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 elds that satisfy these properties, but that such square-root elds always exist.