We propose a conjugate gradient type optimization technique for the computation of the Karcher mean on the set of complex linear subspaces of fixed dimension, modeled by the so-called Grassmannian. The identification of the Grassmannian with Hermitian projection matrices allows an accessible introduction of the geometric concepts required for an intrinsic conjugate gradient method. In particula...

In this paper, we present an accelerated numerical method based on random projection for sparse linear regression. Previous studies have shown that under appropriate conditions, gradient-based methods enjoy a geometric convergence rate when applied to this problem. However, the time complexity of evaluating the gradient is as large as O(nd), where n is the number of data points and d is the dim...

Gradient projection methods based on the Barzilai-Borwein spectral steplength choices are considered for quadratic programming problems with simple constraints. Well known nonmonotone spectral projected gradient methods and variable projection methods are discussed. For both approaches the behavior of different combinations of the two spectral steplengths is investigated. A new adaptive stpleng...

In this paper, a new kind of nonmontone line search method which is called new hybrid projection method with perturbations is proposed. At the same time, global convergence of this kind of method is proved only in the case where the gradient function is uniformly continuous on an open convex set containing the iteration sequence. In doing so, we remove various boundedness conditions. Furthermor...

A new method is introduced for large scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented usi...

Gradient projection methods based on the Barzilai-Borwein spectral steplength choices are considered for quadratic programming problems with simple constraints. Well-known nonmonotone spectral projected gradient methods and variable projection methods are discussed. For both approaches the behavior of different combinations of the two spectral steplengths is investigated. A new adaptive steplen...

This paper deals with unsupervised clustering with feature selection. The problem is to estimate both labels and a sparse projection matrix of weights. To address this combinatorial non-convex problem maintaining a strict control on the sparsity of the matrix of weights, we propose an alternating minimization of the Frobenius norm criterion. We provide a new efficient algorithm named K-sparse w...

The Kelley cutting plane method is one of the methods commonly used to optimize the dual function in the Lagrangian relaxation scheme. Usually the Kelley cutting plane method uses the simplex method as the optimization engine. It is well known that the simplex method leaves the current vertex, follows an ascending edge and stops at the nearest vertex. What would happen if one would continue the...