Linearly Convergent Away-Step Conditional Gradient for Non-strongly Convex Functions

We consider the problem of minimizing a function, which is the sum of a linear function and a composition of a strongly convex function with a linear transformation, over a compact polyhedral set. Jaggi and Lacoste-Julien [14] showed that the conditional gradient method with away steps employed on the aforementioned problem without the additional linear … Read more

The Cyclic Block Conditional Gradient Method for Convex Optimization Problems

In this paper we study the convex problem of optimizing the sum of a smooth function and a compactly supported non-smooth term with a specific separable form. We analyze the block version of the generalized conditional gradient method when the blocks are chosen in a cyclic order. A global sublinear rate of convergence is established … Read more

On the Minimization Over Sparse Symmetric Sets: Projections, Optimality Conditions and Algorithms

We consider the problem of minimizing a general continuously differentiable function over symmetric sets under sparsity constraints. These type of problems are generally hard to solve as the sparsity constraint induces a combinatorial constraint into the problem, rendering the feasible set to be nonconvex. We begin with a study of the properties of the orthogonal … Read more

On the Convergence of Alternating Minimization for Convex Programming with Applications to Iteratively Reweighted Least Squares and Decomposition Schemes

This paper is concerned with the alternating minimization (AM) for solving convex minimization problems where the decision variables vector is split into two blocks. The objective function is a sum of a differentiable convex function and a separable (possible) nonsmooth extended real-valued convex function, and consequently constraints can be incorporated. We analyze the convergence rate … Read more

A First Order Method for Finding Minimal Norm-Like Solutions of Convex Optimization Problems

We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem. We introduce a gradient-based method, called the minimal norm gradient method, for solving this class of problems, and establish … Read more

A Linearly Convergent Algorithm for Solving a Class of Nonconvex/Affine Feasibility Problems

We introduce a class of nonconvex/affine feasibility problems, called (NCF), that consists of finding a point in the intersection of affine constraints with a nonconvex closed set. This class captures some interesting fundamental and NP hard problems arising in various application areas such as sparse recovery of signals and affine rank minimization that we briefly … Read more

Iterative Minimization Schemes for Solving the Single Source Localization Problem

We consider the problem of locating a single radiating source from several noisy measurements using a maximum likelihood (ML) criteria. The resulting optimization problem is nonconvex and nonsmooth and thus finding its global solution is in principal a hard task. Exploiting the special structure of the objective function, we introduce and analyze two iterative schemes … Read more