A Parallel Line Search Subspace Correction Method for Composite Convex Optimization

In this paper, we investigate a parallel subspace correction framework for composite convex optimization. The variables are first divided into a few blocks based on certain rules. At each iteration, the algorithms solve a suitable subproblem on each block simultaneously, construct a search direction by combining their solutions on all blocks, then identify a new … Read more

An Efficient Gauss-Newton Algorithm for Symmetric Low-Rank Product Matrix Approximations

We derive and study a Gauss-Newton method for computing a symmetric low-rank product that is the closest to a given symmetric matrix in Frobenius norm. Our Gauss-Newton method, which has a particularly simple form, shares the same order of iteration-complexity as a gradient method when the size of desired eigenspace is small, but can be … Read more

Trace-Penalty Minimization for Large-scale Eigenspace Computation

The Rayleigh-Ritz (RR) procedure, including orthogonalization, constitutes a major bottleneck in computing relatively high dimensional eigenspaces of large sparse matrices. Although operations involved in RR steps can be parallelized to a certain level, their parallel scalability, which is limited by some inherent sequential steps, is lower than dense matrix-matrix multiplications. The primary motivation of this … Read more

Asset Allocation under the Basel Accord Risk Measures

Financial institutions are currently required to meet more stringent capital requirements than they were before the recent financial crisis; in particular, the capital requirement for a large bank’s trading book under the Basel 2.5 Accord more than doubles that under the Basel II Accord. The significant increase in capital requirements renders it necessary for banks … Read more

Limited Memory Block Krylov Subspace Optimization for Computing Dominant Singular Value Decompositions

In many data-intensive applications, the use of principal component analysis (PCA) and other related techniques is ubiquitous for dimension reduction, data mining or other transformational purposes. Such transformations often require efficiently, reliably and accurately computing dominant singular value decompositions (SVDs) of large unstructured matrices. In this paper, we propose and study a subspace optimization technique … Read more

Uniform bound on the 1-norm of the inverse of lower triangular Toeplitz matrices

The uniform bound of 1-norm is given for the inverse of lower triangular Toeplitz matrices with nonnegative monotonic decreasing entries whose limit is zero. The new bound is the sharpest under the given constraints. This result is then employed to resolve a long standing open problem posed by Brunner concerning the convergence of the one-point … Read more