Simultaneously solving seven optimization problems in relative scale

In this paper we develop and analyze an efficient algorithm which solves seven related optimization problems simultaneously, in relative scale. Each iteration of our method is very cheap, with main work spent on matrix-vector multiplication. We prove that if a certain sequence generated by the algorithm remains bounded, then the method must terminate in $O(1/\delta)$ … Read more

Probing the Pareto frontier for basis pursuit solutions

The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise (BPDN) fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution. We prove that this curve is convex and continuously … Read more

In pursuit of a root

The basis pursuit technique is used to find a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise fits the least-squares problem only approximately, and a single parameter determines a curve that traces the trade-off between the least-squares fit and the one-norm of the solution. We show that the function that describes this … Read more

Coordinate and Subspace Optimization Methods for Linear Least Squares with Non-Quadratic Regularization

This work addresses the problem of regularized linear least squares (RLS) with non-quadratic separable regularization. Despite being frequently deployed in many applications, the RLS problem is often hard to solve using standard iterative methods. In a recent work [10], a new iterative method called Parallel Coordinate Descent (PCD) was devised. We provide herein a convergence … Read more

Sequential Subspace Optimization Method for Large-Scale Unconstrained Problems

We present the Sequential Subspace Optimization (SESOP) method for large scale smooth unconstrained problems. At each iteration we search for a minimum of the objective function over a subspace spanned by the current gradient and by directions of few previous steps. We also include into this subspace the direction from the starting point to the … Read more