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

An Inexact Successive Quadratic Approximation Method for Convex L-1 Regularized Optimization

We study a Newton-like method for the minimization of an objective function that is the sum of a smooth convex function and an l-1 regularization term. This method, which is sometimes referred to in the literature as a proximal Newton method, computes a step by minimizing a piecewise quadratic model of the objective function. In order to make this approach efficient in practice, it is imperative to perform this inner minimization inexactly. In this paper, we give inexactness conditions that guarantee global convergence and that can be used to control the local rate of convergence of the iteration. Our inexactness conditions are based on a semi-smooth function that represents a (continuous) measure of the optimality conditions of the problem, and that embodies the soft-thresholding iteration. We give careful consideration to the algorithm employed for the inner minimization, and report numerical results on two test sets originating in machine learning

Implicit algorithms for eigenvector nonlinearities

We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit viewpoint. More precisely, we change the Newton update equation in a way that the next iterate does not only appear linearly in the update equation. Although, the modifications of the update equation make the methods implicit we show how corresponding iterates can be computed explicitly. Therefore we can carry out steps of the implicit method using explicit procedures. In several cases, these procedures involve a solution of standard eigenvalue problems. We propose two modifications, one of the modifications leads directly to a well-established method (the self-consistent field iteration) whereas the other method is to our knowledge new and has several attractive properties. Convergence theory is provided along with several simulations which illustrate the properties of the algorithms