Related papers: Newton methods beyond Hessian Lipschitz continuity…
Many data-fitting applications require the solution of an optimization problem involving a sum of large number of functions of high dimensional parameter. Here, we consider the problem of minimizing a sum of $n$ functions over a convex…
An adaptive regularization strategy for stabilizing Newton-like iterations on a coarse mesh is developed in the context of adaptive finite element methods for nonlinear PDE. Existence, uniqueness and approximation properties are known for…
An adaptive regularization algorithm using high-order models is proposed for partially separable convexly constrained nonlinear optimization problems whose objective function contains non-Lipschitzian $\ell_q$-norm regularization terms for…
For solving large-scale non-convex problems, we propose inexact variants of trust region and adaptive cubic regularization methods, which, to increase efficiency, incorporate various approximations. In particular, in addition to approximate…
The additive Schwarz method is usually presented as a preconditioner for a PDE linearization based on overlapping subsets of nodes from a global discretization. It has previously been shown how to apply Schwarz preconditioning to a…
The Helmholtz equation poses significant computational challenges due to its oscillatory solutions, particularly for large wavenumbers. Inspired by the Schur complement system for elliptic problems, this paper presents a novel…
The analysis of gradient descent-type methods typically relies on the Lipschitz continuity of the objective gradient. This generally requires an expensive hyperparameter tuning process to appropriately calibrate a stepsize for a given…
In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth…
In this paper, we propose a globally convergent method for solving constrained nonlinear systems. The method combines an efficient Newton conditional gradient method with a derivative-free and nonmonotone linesearch strategy. The global…
An algorithm for solving smooth nonconvex optimization problems is proposed that, in the worst-case, takes $\mathcal{O}(\epsilon^{-3/2})$ iterations to drive the norm of the gradient of the objective function below a prescribed positive…
Frank-Wolfe methods are projection-free algorithms for constrained optimization whose practical performance often depends critically on the choice of step size. Classical closed-loop step-size rules typically require prior knowledge of a…
The numerical solution of partial differential equations on high-dimensional domains gives rise to computationally challenging linear systems. When using standard discretization techniques, the size of the linear system grows exponentially…
We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger…
Machine learning and deep learning are widely researched fields that provide solutions to many modern problems. Due to the complexity of new problems related to the size of datasets, efficient approaches are obligatory. In optimization…
In this paper we study the existence of continuous solutions and their constructions for a second order iterative functional equation, which involves iterate of the unknown function and a nonlinear term. Imposing Lipschitz conditions to…
We study a class of optimization problems on Riemannian manifolds, where the objective function consists of a smooth term and quasi-norm type penalties with exponent $p \in (0, 1]$. The essential difficulty lies in the fact that the…
We present a subgradient method for minimizing non-smooth, non-Lipschitz convex optimization problems. The only structure assumed is that a strictly feasible point is known. We extend the work of Renegar [5] by taking a different…
In this paper, we propose a unified two-phase scheme to accelerate any high-order regularized tensor approximation approach on the smooth part of a composite convex optimization model. The proposed scheme has the advantage of not needing to…
Optimization problems with composite functions consist of an objective function which is the sum of a smooth and a (convex) nonsmooth term. This particular structure is exploited by the class of proximal gradient methods and some of their…
Fixed-point or Newton-methods are typically employed for the numerical solution of nonlinear systems arising from discretization of nonlinear magnetic field problems. We here discuss an alternative strategy which uses local Quasi-Newton…