Interior path following primal-dual algorithms. Part I: Linear programming
Mathematical Programming: Series A and B
On the complexity of following the central path of linear programs by linear extrapolation II
Mathematical Programming: Series A and B - Special issue on interior point methods for linear programming: theory and practice
Applied numerical linear algebra
Applied numerical linear algebra
Linear systems in Jordan algebras and primal-dual interior-point algorithms
Journal of Computational and Applied Mathematics - Special issue: dedicated to William B. Gragg on the occasion of his 60th Birthday
Mathematical Programming: Series A and B
Polynomial Convergence of a New Family of Primal-Dual Algorithms for Semidefinite Programming
SIAM Journal on Optimization
On the Nesterov--Todd Direction in Semidefinite Programming
SIAM Journal on Optimization
On a commutative class of search directions for linear programming over symmetric cones
Journal of Optimization Theory and Applications
Mathematical Programming: Series A and B
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In this paper, we study iteration complexities of Mizuno-Todd-Ye predictor-corrector (MTY-PC) algorithms in SDP and symmetric cone programs by way of curvature integrals. The curvature integral is defined along the central path, reflecting the geometric structure of the central path. Integrating curvature along the central path, we obtain a precise estimate of the number of iterations to solve the problem. It has been shown for LP that the number of iterations is asymptotically precisely estimated with the integral divided by $\sqrt{\beta}$ , where β is the opening parameter of the neighborhood of the central path in MTY-PC algorithms. Furthermore, this estimate agrees quite well with the observed number of iterations of the algorithm even when β is close to one and when applied to solve large LP instances from NETLIB. The purpose of this paper is to develop direct extensions of these two results to SDP and symmetric cone programs. More specifically, we give concrete formulas for curvature integrals in SDP and symmetric cone programs and give asymptotic estimates for iteration complexities. Through numerical experiments with large SDP instances from SDPLIB, we demonstrate that the number of iterations is explained quite well with the integral even for a large step size which is enough to solve practical large problems.