Convergence analysis of an Inexact Infeasible Interior Point method for Semidefinite Programming

In this paper we present an extension to SDP of the well known infeasible Interior Point method for linear programming of Kojima,Megiddo and Mizuno (A primal-dual infeasible-interior-point algorithm for Linear Programming, Math. Progr., 1993). The extension developed here allows the use of inexact search directions; i.e., the linear systems defining the search directions can be solved with an accuracy that increases as the solution is approached. A convergence analysis is carried out and the global convergence of the method is prove

Local quadratic convergence of polynomial-time interior-point methods for conic optimization problems

In this paper, we establish a local quadratic convergence of polynomial-time interior-point methods for general conic optimization problems. The main structural property used in our analysis is the logarithmic homogeneity of self-concordant barrier functions. We propose new path-following predictor-corrector schemes which work only in the dual space. They are based on an easily computable gradient proximity measure, which ensures an automatic transformation of the global linear rate of convergence to the local quadratic one under some mild assumptions. Our step-size procedure for the predictor step is related to the maximum step size (the one that takes us to the boundary). It appears that in order to obtain local superlinear convergence, we need to tighten the neighborhood of the central path proportionally to the current duality gapconic optimization problem, worst-case complexity analysis, self-concordant barriers, polynomial-time methods, predictor-corrector methods, local quadratic convergence

Asymptotic behavior of underlying NT paths in interior point methods for monotone semidefinite linear complementarity problems

2010-2011 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

Behavioral Measures and their Correlation with IPM Iteration Counts on Semi-Definite Programming Problems

We study four measures of problem instance behavior that might account for the observed differences in interior-point method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar-) condition measure C(d) of the data instance, (iii) a measure of the near-absence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the correlation between these measures and IPM iteration counts (solved using the software SDPT3) when the measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.896), and is a very good predictor of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The near-absence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations

Superlinear convergence of an infeasible predictor-corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg-Kojima-Monteiro direction

2010-2011 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Local Convergence of Predictor-Corrector Infeasible-Interior-Point Algorithms for SDPs and SDLCPs

. An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the Mizuno-Todd-Ye type predictorcorrector primal-dual interior-point method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A Mizuno-Todd-Ye type predictor-corrector infeasible-interior-point algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. Key words. Semidefinite Programming, Infeasible-Interior-Point Method, Predictor-CorrectorMethod, Superlinear Convergence, Primal-Dual Nondegeneracy Abbreviated Title. Interior-Point Algorithms for SDPs y Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japa..