Dual Randomized Coordinate Descent Method for Solving a Class of Nonconvex Problems

We consider a nonconvex optimization problem consisting of maximizing the difference of two convex functions. We present a randomized method that requires low computational effort at each iteration. The described method is a randomized coordinate descent method employed on the so-called Toland-dual problem. We prove subsequence convergence to dual stationarity points, a new notion that … Read more

On the Convergence to Stationary Points of Deterministic and Randomized Feasible Descent Directions Methods

This paper studies the class of nonsmooth nonconvex problems in which the difference between a continuously differentiable function and a convex nonsmooth function is minimized over linear constraints. Our goal is to attain a point satisfying the stationarity necessary optimality condition, defined as the lack of feasible descent directions. Although elementary in smooth optimization, this … Read more

On the Convergence to Stationary Points of Deterministic and Randomized Feasible Descent Directions Methods

We study the class of nonsmooth nonconvex problems in which the objective is to minimize the difference between a continuously differentiable function (possibly nonconvex) and a convex (possibly nonsmooth) function over a convex polytope. This general class contains many types of problems, including difference of convex functions (DC) problems, and as such, can be used … Read more

Globally Solving the Trust Region Subproblem Using Simple First-Order Methods

We consider the trust region subproblem which is given by a minimization of a quadratic, not necessarily convex, function over the Euclidean ball. Based on the well-known second-order necessary and sufficient optimality conditions for this problem, we present two sufficient optimality conditions defined solely in terms of the primal variables. Each of these conditions corresponds … Read more

FOM — A MATLAB Toolbox of First Order Methods for Solving Convex Optimization Problems

This paper presents the FOM MATLAB toolbox for solving convex optimization problems using first order methods. The diverse features of the eight solvers included in the package are illustrated through a collection of examples of different nature. Article Download View FOM — A MATLAB Toolbox of First Order Methods for Solving Convex Optimization Problems

Globally Solving a Class of Optimal Power Flow Problems in Radial Networks by Tree Reduction

We devise an algorithm for finding the global optimal solution of the so-called optimal power flow problem (OPF) for a class of power networks with a tree topology, also called radial networks, for which an efficient and reliable algorithm was not previously known. The algorithm we present is called the tree reduction/expansion method, and is … Read more

Proximal Mapping for Symmetric Penalty and Sparsity

This paper studies a class of problems consisting of minimizing a continuously differentiable function penalized with the so-called $\ell_0$-norm over a symmetric set. These problems are hard to solve, yet prominent in many fields and applications. We first study the proximal mapping with respect to the $\ell_0$-norm over symmetric sets, and provide an efficient method … Read more

A Branch and Bound Algorithm for Nonconvex Quadratic Optimization with Ball and Linear Constraints

We suggest a branch and bound algorithm for solving continuous optimization problems where a (generally nonconvex) objective function is to be minimized under nonconvex inequality constraints which satisfy some specific solvability assumptions. The assumptions hold for some special cases of nonconvex quadratic optimization problems. We show how the algorithm can be applied to the problem … Read more

The Sparse PCA Problem: Optimality Conditions and Algorithms

Sparse principal component analysis (PCA) addresses the problem of finding a linear combination of the variables in a given data set with a sparse coefficients vector that maximizes the variability of the data. This model enhances the ability to interpret the principal components, and is applicable in a wide variety of fields including genetics and … Read more