Semi-Algebraic Structure and Central Paths in Nonlinear Optimization

We consider the concept of so-called central path in nonlinear optimization that lies at the heart of interior point methods. Using singularity theory and real algebraic geometry, we provide conditions for the existence, convergence, and analyticity of central paths of nonlinear optimization problems. We also prove quantitative bounds on the convergence rate and analyticity of the central path for the special case of semi-definite optimization.