FISTA and Extensions – Review and New Insights

The purpose of this technical report is to review the main properties of an accelerated composite gradient (ACG) method commonly referred to as the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA). In addition, we state a version of FISTA for solving both convex and strongly convex composite minimization problems and derive its iteration complexities to generate iterates … Read more

Analysis of the Frank-Wolfe Method for Convex Composite Optimization involving a Logarithmically-Homogeneous Barrier

We present and analyze a new generalized Frank-Wolfe method for the composite optimization problem (P): F*:= min_x f(Ax) + h(x), where f is a \theta-logarithmically-homogeneous self-concordant barrier and the function h has bounded domain but is possibly non-smooth. We show that our generalized Frank-Wolfe method requires O((Gap_0 + \theta + Var_h)\ln(\delta_0) + (\theta + Var_h)^2/\epsilon) … Read more

A Proximal Quasi-Newton Trust-Region Method for Nonsmooth Regularized Optimization

We develop a trust-region method for minimizing the sum of a smooth term f and a nonsmooth term h, both of which can be nonconvex. Each iteration of our method minimizes apossibly nonconvex model of f+h in a trust region. The model coincides with f+h in value and subdifferential at the center. We establish global … Read more

Beyond Alternating Updates for Matrix Factorization with Inertial Bregman Proximal Gradient Algorithms

Matrix Factorization is a popular non-convex objective, for which alternating minimization schemes are mostly used. They usually suffer from the major drawback that the solution is biased towards one of the optimization variables. A remedy is non-alternating schemes. However, due to a lack of Lipschitz continuity of the gradient in matrix factorization problems, convergence cannot … Read more

Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence

The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically defined, scales poorly with the dimension of the ambient space. In contrast, we here show that in a variety of concrete circumstances, … Read more

Convex-Concave Backtracking for Inertial Bregman Proximal Gradient Algorithms in Non-Convex Optimization

Backtracking line-search is an old yet powerful strategy for finding better step size to be used in proximal gradient algorithms. The main principle is to locally find a simple convex upper bound of the objective function, which in turn controls the step size that is used. In case of inertial proximal gradient algorithms, the situation … Read more

Composite optimization for robust blind deconvolution

The blind deconvolution problem seeks to recover a pair of vectors from a set of rank one bilinear measurements. We consider a natural nonsmooth formulation of the problem and show that under standard statistical assumptions, its moduli of weak convexity, sharpness, and Lipschitz continuity are all dimension independent. This phenomenon persists even when up to … Read more

Golden Ratio Algorithms for Variational Inequalities

The paper presents a fully explicit algorithm for monotone variational inequalities. The method uses variable stepsizes that are computed using two previous iterates as an approximation of the local Lipschitz constant without running a linesearch. Thus, each iteration of the method requires only one evaluation of a monotone operator $F$ and a proximal mapping $g$. … Read more

Efficiency of minimizing compositions of convex functions and smooth maps

We consider the problem of minimizing a sum of a convex function and a composition of a convex function with a smooth map. Important examples include exact penalty formulations of nonlinear programs and nonlinear least squares problems with side constraints. The basic algorithm we rely on is the well-known prox-linear method, which in each iteration … Read more

The Rate of Convergence of Augmented Lagrange Method for a Composite Optimization Problem

In this paper we analyze the rate of local convergence of the augmented Lagrange method for solving optimization problems with equality constraints and the objective function expressed as the sum of a convex function and a twice continuously differentiable function. The presence of the non-smoothness of the convex function in the objective requires extensive tools … Read more