Information geometry and interior-point algorithms

Abstract

In this paper, we introduce a geometric theory which relates a geometric structure of convex optimization problems to computational complexity to solve the problems. Specifically, we develop information geometric framework of conic linear optimization problems and show that the iteration complexity of the standard polynomial-time primal-dual predictor-corrector interior-point algorithms to solve symmetric cone programs is written with an information geometric curvature integral of the central path which the algorithm traces. Numerical experiments demonstrate that the number of iterations is quite well explained with the integral even for the large problems with thousands of variables; we claim that the iteration-complexity of the primal-dual predictor-corrector path-following algorithm is an information geometric quantity. We also develop a global theorem about the central path for linear programs. © 2013 Springer-Verlag.