We propose and analyse primal-dual interior-point algorithms for convex optimization problems in conic form. The families of algorithms whose iteration complexity we analyse are so-called short-step algorithms. Our iteration complexity bounds match the current best iteration complexity bounds for primal-dual symmetric interior-point algorithm of Nesterov and Todd, for symmetric cone programming...

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 this paper we show that the so-called commutative class of primal-dual interior point algorithms which were designed by Monteiro and Zhang for semidefinite programming extends word-for-word to optimization problems over all symmetric cones. The machinery of Euclidean Jordan algebras is used to carry out this extension. Unlike some non-commutative algorithms such as the XS+SXmethod, this clas...

In this paper, we introduce the truncated primal-infeasible dual-feasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum cost network flow problem. In each iteration, the linear system that determines the search direction is computed inexactly, and the norm of the resulting residual vector is used in the stopping criteria...

Recently, infeasibility issues in interior methods for nonconvex nonlinear programming have been studied. In particular, it has been shown how many line-search interior methods may converge to an infeasible point which is on the boundary of the feasible region with respect to the inequality constraints. The convergence is such that the search direction does not tend to zero, but the step length...

This paper proposes a new predictor-corrector interior-point method for a class of semidefinite programs, which numerically traces the central trajectory in a space of Lagrange multipliers. The distinguished features of the method are full use of the BFGS quasi-Newton method in the corrector procedure and an application of the conjugate gradient method with an effective preconditioning matrix i...

In this paper, we propose a feasible primal-dual path-following algorithm for convex quadratic programs.At each interior-point iteration the algorithm uses a full-Newton step and a suitable proximity measure for tracing approximately the central path.We show that the short-step algorithm has the best known iteration bound,namely O( √ n log (n+1) ).

We propose a family of search directions based on primal-dual entropy in the contextof interior-point methods for linear optimization. We show that by using entropy based searchdirections in the predictor step of a predictor-corrector algorithm together with a homogeneousself-dual embedding, we can achieve the current best iteration complexity bound for linear opti-mization. The...

In this paper we present a specialized matrix factorization procedure for computing the dual step in a primal-dual path-following interior point algorithm for solving two-stage stochastic linear programs with restricted recourse. The algorithm, based on the Birge-Qi factorization technique, takes advantage of both the dual block-angular structure of the constraint matrix and of the special stru...

Interior Point Stabilization (IPS) is an acceleration method for Column Generation algorithms. Like previous stabilization techniques, it addresses degeneracy problems and convergence difficulties by preventing dual variables from taking extreme values. IPS is however different since it selects a dual solution inside the optimal dual space rather than retrieving an extreme point solution from t...