In this paper, we study polynomial-time interior-point algorithms in view of information geometry. Information geometry is a differential geometric framework which has been successfully applied to statistics, learning theory, signal processing etc. We consider information geometric structure for conic linear programs introduced by self-concordant barrier functions, and develop a precise iterati...

In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semideenite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semideenite matrix completion, and it can be embodied in two diierent ways. One is by a con...

In this paper we present a generic primal-dual interior point methods (IPMs) for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. The proposed kernel function does not satisfy all the conditions proposed in [2]. We show that the corresponding large-update algorithm improves the ite...

In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two different ways. One is by a ...

Interior-point methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixed-integer nonlinear programming problems via outer approximation. However, traditionally, infeasible primal-dual interior-point methods have had two main...

The matrix variables in a primal-dual pair of semidefinite programs are getting increasingly ill-conditioned as they approach a complementary solution. Multiplying the primal matrix variable with a vector from the eigenspace of the non-basic part will therefore result in heavy numerical cancellation. This effect is amplified by the scaling operation in interior point methods. A complete example...

We present primal-dual interior-point algorithms with polynomial iteration bounds to nd approximate solutions of semidenite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly improved.

In [H. Mansouri and C. Roos, Numer. Algorithms 52 (2009) 225-255.], Mansouri and Ross presented a primal-dual infeasible interior-point algorithm with full-Newton steps whose iteration bound coincides with the best known bound for infeasible interior-point methods. Here, we introduce a slightly different algorithm with a different search direction and show that the same complexity result is obt...

In this paper we extend the results obtained for a class of finite kernel functions by Y.Q. Bai M. El Ghami and C.Roos published in SIAM Journal of Optimization, 13(3):766–782, 2003 [3] for linear optimization to semidefinite optimization. We show that the iteration bound for primal dual methods is O( √ n log n log n ), for large-update methods andO( √ n log n ), for small-update methods. The i...

We introduce a new barrier function which has a linear growth term in its kernel function. So far all existing kernel functions have a quadratic (or higher degree) growth term. Despite this, a large-update primal-dual interior-point method based on this kernel function has the same iteration bound as the classical primal-dual method, which is based on the logarithmic barrier method.