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We introduce a notion of inexact model of a convex objective function, which allows for errors both in the function and in its gradient. For this situation, a gradient method with an adaptive adjustment of some parameters of the model is…
We establish or refute the optimality of inexact second-order methods for unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing the results of Cartis, Gould and Toint…
We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate…
This study addresses some algorithms for solving structured unconstrained convex optimiza- tion problems using first-order information where the underlying function includes high-dimensional data. The primary aim is to develop an…
In this paper, a globally convergent Newton-type proximal gradient method is developed for composite multi-objective optimization problems where each objective function can be represented as the sum of a smooth function and a nonsmooth…
We consider the problem of minimizing a convex objective which is the sum of a smooth part, with Lipschitz continuous gradient, and a nonsmooth part. Inspired by various applications, we focus on the case when the nonsmooth part is a…
We present a simple transformation of any linear program or semidefinite program into an equivalent convex optimization problem whose only constraints are linear equations. The objective function is defined on the whole space, making…
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…
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…
In this paper some adaptive mirror descent algorithms for problems of minimization convex objective functional with several convex Lipschitz (generally, non-smooth) functional constraints are considered. It is shown that the methods are…
We investigate a class of composite nonconvex functions, where the outer function is the sum of univariate extended-real-valued convex functions and the inner function is the limit of difference-of-convex functions. A notable feature of…
In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz…
In this paper, we consider an unconstrained stochastic optimization problem where the objective function exhibits high-order smoothness. Specifically, we propose a new stochastic first-order method (SFOM) with multi-extrapolated momentum,…
This paper optimizes the step coefficients of first-order methods for smooth convex minimization in terms of the worst-case convergence bound (i.e., efficiency) of the decrease in the gradient norm. This work is based on the performance…
Composite minimization involves a collection of functions which are aggregated in a nonsmooth manner. It covers, as a particular case, smooth approximation of minimax games, minimization of max-type functions, and simple composite…
This paper initiates a systematic development of a theory of non-commutative optimization. It aims to unify and generalize a growing body of work from the past few years which developed and analyzed algorithms for natural geodesically…
This paper presents a unified analysis for the proximal subgradient method (Prox-SubGrad) type approach to minimize an overall objective of $f(x)+r(x)$, subject to convex constraints, where both $f$ and $r$ are weakly convex, nonsmooth, and…
This paper presents a stochastic block-coordinate proximal Newton method for minimizing the sum of a blockwise Lipschitz-continuously differentiable function and a separable nonsmooth convex function. At each iteration, the method randomly…
We consider solving nonconvex composite optimization problems in which the sum of a smooth function and a nonsmooth function is minimized. Many of convergence analyses of proximal gradient-type methods rely on global descent property…
In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many…