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We present two approximate versions of the proximal subgradient method for minimizing the sum of two convex functions (not necessarily differentiable). The algorithms involve, at each iteration, inexact evaluations of the proximal operator…
We introduce an efficient first-order primal-dual method for the solution of nonsmooth PDE-constrained optimization problems. We achieve this efficiency through not solving the PDE or its linearisation on each iteration of the optimization…
This paper addresses a class of general nonsmooth and nonconvex composite optimization problems subject to nonlinear equality constraints. We assume that a part of the objective function and the functional constraints exhibit local…
This paper is devoted to the study of an inertial accelerated primal-dual algorithm, which is based on a second-order differential system with time scaling, for solving a non-smooth convex optimization problem with linear equality…
Recent advances in convex optimization have leveraged computer-assisted proofs to develop optimized first-order methods that improve over classical algorithms. However, each optimized method is specially tailored for a particular problem…
In this paper, we introduce faster accelerated primal-dual algorithms for minimizing a convex function subject to strongly convex function constraints. Prior to our work, the best complexity bound was $\mathcal{O}(1/{\varepsilon})$,…
In this paper we consider non-smooth convex optimization problems with (possibly) infinite intersection of constraints. In contrast to the classical approach, where the constraints are usually represented as intersection of simple sets,…
This paper seeks to address how to solve non-smooth convex and strongly convex optimization problems with functional constraints. The introduced Mirror Descent (MD) method with adaptive stepsizes is shown to have a better convergence rate…
We study the non-smooth optimization problems in machine learning, where both the loss function and the regularizer are non-smooth functions. Previous studies on efficient empirical loss minimization assume either a smooth loss function or…
The performance of optimization methods is often tied to the spectrum of the objective Hessian. Yet, conventional assumptions, such as smoothness, do often not enable us to make finely-grained convergence statements -- particularly not for…
We design accelerated algorithms with improved rates for several fundamental classes of optimization problems. Our algorithms all build upon techniques related to the analysis of primal-dual extragradient methods via relative Lipschitzness…
We consider a general multi-agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint, a global equality constraint, and a global constraint set.…
The Primal-Dual hybrid gradient (PDHG) method is a powerful optimization scheme that breaks complex problems into simple sub-steps. Unfortunately, PDHG methods require the user to choose stepsize parameters, and the speed of convergence is…
In this paper we propose distributed dual gradient algorithms for linearly constrained separable convex problems and analyze their rate of convergence under different assumptions. Under the strong convexity assumption on the primal…
In this paper, we deal with multiobjective composite optimization problems, where each objective function is a combination of smooth and possibly non-smooth functions. We first propose a parameter-dependent conditional gradient method to…
We analyze several generic proximal splitting algorithms well suited for large-scale convex nonsmooth optimization. We derive sublinear and linear convergence results with new rates on the function value suboptimality or distance to the…
This paper proposes a new steepest gradient descent method for solving nonconvex finite minimax problems using non-monotone adaptive step sizes and providing proof of convergence results in cases of the nonconvex, quasiconvex, and…
We propose a new subgradient method for the minimization of nonsmooth convex functions over a convex set. To speed up computations we use adaptive approximate projections only requiring to move within a certain distance of the exact…
Binary optimization is a powerful tool for modeling combinatorial problems, yet scalable and theoretically sound solution methods remain elusive. Conventional solvers often rely on heuristic strategies with weak guarantees or struggle with…
Best subset selection is considered the `gold standard' for many sparse learning problems. A variety of optimization techniques have been proposed to attack this non-smooth non-convex problem. In this paper, we investigate the dual forms of…