We present a new hybrid algorithm based on Godunov’s method for computing eigenvectors of symmetric tridiagonal matrices and Inverse Iteration, which we call the Godunov–Inverse Iteration. We use eigenvectors computed according to Godunov’s method as starting vectors in the Inverse Iteration, replacing any nonnumeric elements of Godunov’s eigenvectors with random uniform numbers. We use the rig...

We propose and study the iteration-complexity of a proximal-Newton method for finding approximate solutions of the problem of minimizing a twice continuously differentiable convex function on a (possibly infinite dimensional) Hilbert space. We prove global convergence rates for obtaining approximate solutions in terms of function/gradient values. Our main results follow from an iteration-comple...

We propose a modified alternate direction method for solving convex quadratically constrained quadratic semidefinite optimization problems. The method is a first-order method, therefore requires much less computational effort per iteration than the second-order approaches such as the interior point methods or the smoothing Newton methods. In fact, only a single inexact metric projection onto th...

The multivariate eigenvalue problem (MEP) which originally arises from the canonical correlation analysis is an important generalization of the classical eigenvalue problem. Recently, the MEP also finds applications in many other areas and continues to receive interest. However, the existing algorithms for the MEP are the generalization of the power iteration for the classical eigenvalue proble...

The paper studies the solution of stochastic optimization problems in which approximations to the gradient and Hessian are obtained through subsampling. We first consider Newton-like methods that employ these approximations and discuss how to coordinate the accuracy in the gradient and Hessian to yield a superlinear rate of convergence in expectation. The second part of the paper analyzes an in...

In this paper, the authors present and analyze a new hybrid inexact Logarithmic-Quadratic Proximal method for solving nonlinear complementarity problems. Each iteration of the new method consists of a prediction and a correction step. The predictor is produced using an inexact Logarithmic-Quadratic Proximal method, which is then corrected by the Proximal Point Algorithm. The new iterate is obta...