On the complexity of analyticity in semi-definite optimization

It is well-known that the central path of semi-definite optimization, unlike linear optimization, has no analytic extension to $\mu =0$ in the absence of the strict complementarity condition. In this paper, we consider a reparametrization $\mu \mapsto {\mu }^{\rho }$, with ρ being a positive integer, that recovers the analyticity of the central path at $\mu =0$. We investigate the complexity of computing ρ using algorithmic real algebraic geometry and the theory of complex algebraic curves. We prove that the optimal ρ is bounded by $2^{O(m^2+n^{2}m+n^4)}$, where n is the matrix size and m is the number of affine constraints. Our approach leads to a symbolic algorithm, based on the Newton-Puiseux algorithm, which computes a feasible ρ using $2^{O(m+n^2)}$ arithmetic operations.