In this paper, we present a primal-dual interior-point method for solving nonlinear programming problems. It employs a Levenberg-Marquardt (LM) perturbation to the Karush-Kuhn-Tucker (KKT) matrix to handle indefinite Hessians and a line search to obtain sufficient descent at each iteration. We show that the LM perturbation is equivalent to replacing the Newton step by a cubic regularization ste...

We discuss fast implementations of primal-dual interior-point methods for semidefinite programs derived from the Kalman-Yakubovich-Popov lemma, a class of problems that are widely encountered in control and signal processing applications. By exploiting problem structure we achieve a reduction of the complexity by several orders of magnitude compared to generalpurpose semidefinite programming so...

The Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro and the Nesterov-Todd search directions have been used in many primal-dual interior-point methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when a semidefinite program to be solved is large scale and sparse.

The Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro and the Nesterov-Todd search directions have been used in many primal-dual interior-point methods for semide nite programs. This paper proposes an e cient method for computing the two directions when a semide nite program to be solved is large scale and sparse.

The theory of self-scaled conic programming provides a uniied framework for the theories of linear programming, semideenite programming and convex quadratic programming with convex quadratic constraints. The standard search directions for interior-point methods applied to self-scaled conic programming problems are the so-called Nesterov-Todd directions. In this article we show that these direct...

We propose a family of directions that generalizes many directions proposed so far in interiorpoint methods for the SDP (semide nite programming) and for the monotone SDLCP (semide nite linear complementarity problem). We derive the family from the Helmberg-Rendl-Vanderbei-Wolkowicz/KojimaShindoh-Hara/Monteiro direction by relaxing its \centrality equation" into a \centrality inequality." Using...

We characterize the spectral behavior of a primal Schur-complement-based block diagonal preconditioner for saddle point systems, subject to low-rank modifications. This is motivated by a desire to reduce as much as possible the computational cost of matrix-vector products with the (1,1) block, while keeping the eigenvalues of the preconditioned matrix reasonably clustered. The formulation leads...

The rediscovery of the affine scaling method in the late 80s was one of the turning points which led to a new chapter in Modern Optimization the Interior Point Methods (IPMs). The purpose of this paper is to show the intrinsic connections between Interior and Exterior Point methods (EPMs), which have been developed during the last 30 years. A class Ψ of smooth and strictly concave functions ψ :...

Kernel functions play an important role in the complexity analysis of interior point methods (IPMs) for linear optimization (LO). In this paper, interior-point algorithm LO based on a new parametric kernel function is proposed. By means some simple tools, we prove that primal-dual solving problems meets $ O\left(\sqrt{n} \log(n) \log(\frac{n}{\varepsilon}) \right) $, iteration bound large-updat...

A class of primal dual interior point methods is developed These methods use a higher order Taylor polynomial to approximate a primal dual trajectory de ned from an infeasible point A detailed implementation of these methods is given and their computational performance is studied We give computational results for two di erent approaches for de ning trajectory and using up to th order polynomial...