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We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal…
Natural Gradient Descent (NGD) has emerged as a promising optimization algorithm for training neural network-based solvers for partial differential equations (PDEs), such as Physics-Informed Neural Networks (PINNs). However, its practical…
We introduce a novel approach addressing global analysis of a difficult class of nonconvex-nonsmooth optimization problems within the important framework of Lagrangian-based methods. This genuine nonlinear class captures many problems in…
It is widely accepted that the stepsize is of great significance to gradient method. Two efficient gradient methods with approximately optimal stepsizes mainly based on regularization models are proposed for unconstrained optimization. More…
Line-search methods are commonly used to solve optimization problems. The simplest line search method is steepest descent where one always moves in the direction of the negative gradient. Newton's method on the other hand is a second-order…
Solving structured systems of linear equations in a non-centralized fashion is an important step in many distributed optimization and control algorithms. Fast convergence is required in manifold applications. Known decentralized algorithms,…
In this work we develop a numerical method for solving a type of convex graph-structured tensor optimization problems. This type of problems, which can be seen as a generalization of multi-marginal optimal transport problems with…
Optimization techniques play a crucial role in estimating parameters and state information for nonlinear systems. However, some critical aspects of these problems have received little attention in previous research. In this paper, we…
We develop multi-step gradient methods for network-constrained optimization of strongly convex functions with Lipschitz-continuous gradients. Given the topology of the underlying network and bounds on the Hessian of the objective function,…
$L^1$ based optimization is widely used in image denoising, machine learning and related applications. One of the main features of such approach is that it naturally provide a sparse structure in the numerical solutions. In this paper, we…
This paper deals with nonsmooth convex optimization problems in Euclidean spaces. We identify special elements of the subdifferential of a convex function, called specular gradients. Based on this observation, we propose three numerical…
We propose a variant of the classical conditional gradient method for sparse inverse problems with differentiable measurement models. Such models arise in many practical problems including superresolution, time-series modeling, and matrix…
This paper studies the numerical solution of strictly convex unconstrained optimization problems by linesearch Newton-CG methods. We focus on methods employing inexact evaluations of the objective function and inexact and possibly random…
We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton's method and the linear conjugate gradient algorithm, with explicit detection and use of negative curvature directions for the…
Despite the established convergence theory of Optimistic Gradient Descent Ascent (OGDA) and Extragradient (EG) methods for the convex-concave minimax problems, little is known about the theoretical guarantees of these methods in nonconvex…
The Conjugate Gradient method (CGM) is known to be the fastest generic iterative method for solving linear systems with symmetric sign definite matrices. In this paper, we modify this method so that it could find fundamental solitary waves…
In this paper, we explore two fundamental first-order algorithms in convex optimization, namely, gradient descent (GD) and proximal gradient method (ProxGD). Our focus is on making these algorithms entirely adaptive by leveraging local…
We consider a class of (possibly strongly) geodesically convex optimization problems on Hadamard manifolds, where the objective function splits into the sum of a smooth and a possibly nonsmooth function. We introduce an intrinsic convex…
In structural optimization, both parameters and shape are relevant for the model performance. Yet, conventional optimization techniques usually consider either parameters or the shape separately. This work addresses this problem by…
The goal of this paper is to reduce the total complexity of gradient-based methods for two classes of problems: affine-constrained composite convex optimization and bilinear saddle-point structured non-smooth convex optimization. Our…