Numerical Investigation of Crouzeix’s Conjecture
Crouzeix’s conjecture states that for all polynomials p and matrices A, the inequality ||p(A)||
Nonsmooth Optimization
Homotopy Smoothing for Non-Smooth Problems with Lower Complexity than O(1/epsilon)
In this paper, we develop a novel {\bf ho}moto{\bf p}y {\bf s}moothing (HOPS) algorithm for solving a family of non-smooth problems that is composed of a non-smooth term with an explicit max-structure and a smooth term or a simple non-smooth term whose proximal mapping is easy to compute. The best known iteration complexity for solving … Read more
A general double-proximal gradient algorithm for d.c. programming
The possibilities of exploiting the special structure of d.c. programs, which consist of optimizing the difference of convex functions, are currently more or less limited to variants of the DCA proposed by Pham Dinh Tao and Le Thi Hoai An in 1997. These assume that either the convex or the concave part, or both, are … Read more
Nonsmooth optimization using Taylor-like models: error bounds, convergence, and termination criteria
We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our main result is an explicit relationship between the step-size of any such algorithm and the slope of the function at a … Read more
A SMART Stochastic Algorithm for Nonconvex Optimization with Applications to Robust Machine Learning
Machine learning theory typically assumes that training data is unbiased and not adversarially generated. When real training data deviates from these assumptions, trained models make erroneous predictions, sometimes with disastrous effects. Robust losses, such as the huber norm are designed to mitigate the effects of such contaminated data, but they are limited to the regression … Read more
On the convergence of a regularized Jacobi algorithm for convex optimization
In this paper we consider the regularized version of the Jacobi algorithm, a block coordinate descent method for convex optimization with differentiable objective function and block-separable constraints that has been recently proposed in the literature. Under certain regularity assumptions on the objective function, this algorithm has been shown to satisfy the so-called sufficient decrease condition, … Read more
A first-order primal-dual algorithm with linesearch
The paper proposes a linesearch for the primal-dual method. Each iteration of the linesearch requires to update only the dual (or primal) variable. For many problems, in particular for regularized least squares, the linesearch does not require any additional matrix-vector multiplications. We prove convergence of the proposed method under the standard assumptions. We also show … Read more
A highly efficient semismooth Newton augmented Lagrangian method for solving Lasso problems
We develop a fast and robust algorithm for solving large scale convex composite optimization models with an emphasis on the $\ell_1$-regularized least squares regression (Lasso) problems. Despite the fact that there exist a large number of solvers in the literature for the Lasso problems, we found that no solver can efficiently handle difficult large scale … Read more
A progressive barrier derivative-free trust-region algorithm for constrained optimization
We study derivative-free constrained optimization problems and propose a trust-region method that builds linear or quadratic models around the best feasible and and around the best infeasible solutions found so far. These models are optimized within a trust region, and the progressive barrier methodology handles the constraints by progressively pushing the infeasible solutions toward the … Read more
Exact Worst-case Performance of First-order Methods for Composite Convex Optimization
We provide a framework for computing the exact worst-case performance of any algorithm belonging to a broad class of oracle-based first-order methods for composite convex optimization, including those performing explicit, projected, proximal, conditional and inexact (sub)gradient steps. We simultaneously obtain tight worst-case guarantees and explicit instances of optimization problems on which the algorithm reaches this … Read more