Primal-dual Interior-Point Methods (IPMs) have shown their ability in solving large classes of optimization problems efficiently. Feasible IPMs require a strictly feasible starting point to generate the iterates that converge to an optimal solution. The self-dual embedding model provides an elegant solution to this problem with the cost of slightly increasing the size of the problem. On the oth...

In each iteration of an interior-point method for semideenite programming, the maximum step-length that can be taken by the iterate while maintaining the positive semideeniteness constraint need to be estimated. In this note, we show how the maximum step-length can be estimated via the Lanczos iteration, a standard iterative method for estimating the extremal eigenvalues of a matrix. We also gi...

A modiication of the (infeasible) primal{dual interior point method is developed. The method uses multiple corrections to improve the centrality of the current iterate. The maximum number of corrections the algorithm is encouraged to make depends on the ratio of the eeorts to solve and to factorize the KKT systems. For any LP problem, this ratio is determined right after preprocessing the KKT s...

This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming obtained from centrality equations of the form XS + SX = 2νW , where W is a fixed positive definite matrix and ν > 0 is a parameter, under the assumption that the problem has a strictly complementary primal–dual optimal solution. We present a different and simpler proof than the one given ...

We consider the problem of !nding a linear iteration that yields distributed averaging consensus over a network, i.e., that asymptotically computes the average of some initial values given at the nodes. When the iteration is assumed symmetric, the problem of !nding the fastest converging linear iteration can be cast as a semide!nite program, and therefore e"ciently and globally solved. These op...

A linear program has a unique least 2-norm solution provided that the linear program has a solution. To locate this solution, most of the existing methods were devised to solve certain equivalent perturbed quadratic programs or unconstrained minimization problems. Different from these traditional methods, we provide in this paper a new theory and an effective numerical method to seek the least ...

In this paper we present a convergence analysis for some inexact (polynomial) variants of the infeasible-interior-point-algorithm of Kojima, Megiddo and Mizuno. For this analysis we assume that the iterates are bounded. The new variants allow the use of search directions that are calculated from the deening linear system with only moderate accuracy, e.g. via the use of Krylov subspace methods l...

Several algorithms in optimization can be viewed as following a solution as a parameter or set of parameters is adjusted to a desired value. Examples include homotopy methods in complementarity problems and path-following (infeasible-) interior-point methods. If we have a metric in solution space that corresponds to the complexity of moving from one solution point to another, there is an induce...