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We study the convergence properties of a general inertial first-order proximal splitting algorithm for solving nonconvex nonsmooth optimization problems. Using the Kurdyka--\L ojaziewicz (KL) inequality we establish new convergence rates…

Optimization and Control · Mathematics 2016-09-14 Patrick R. Johnstone , Pierre Moulin

In this technical note, we are concerned with the problem of solving variational inequalities with improved convergence rates. Motivated by Nesterov's accelerated gradient method for convex optimization, we propose a Nesterov's accelerated…

Optimization and Control · Mathematics 2022-12-21 Shaolin Tan , Jinhu Lu

The purpose of this paper is to establish the almost sure weak ergodic convergence of a sequence of iterates $(x_n)$ given by $x_{n+1} = (I+\lambda_n A(\xi_{n+1},\,.\,))^{-1}(x_n)$ where $(A(s,\,.\,):s\in E)$ is a collection of maximal…

Optimization and Control · Mathematics 2016-07-26 Pascal Bianchi

For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. % By exploiting…

Numerical Analysis · Mathematics 2015-05-20 Qinian Jin , Ulrich Tautenhahn

In this paper we focus on the convergence analysis of the forward-backward splitting method for solving nonsmooth optimization problems in Hilbert spaces when the objective function is the sum of two convex functions. Assuming that one of…

Optimization and Control · Mathematics 2016-10-17 J. Y. Bello Cruz , T. T. A. Nghia

In this paper we study the pseudomonotone equilibrium problem. We consider a new inertial condition for the subgradient extragradient method with self-adaptive step size for approximating a solution of the equilibrium problem in a real…

Optimization and Control · Mathematics 2023-11-01 Chinedu Izuchukwu , Grace Ogwo , Bertin Zinsou

We define a stochastic variant of the proximal point algorithm in the general setting of nonlinear (separable) Hadamard spaces for approximating zeros of the mean of a stochastically perturbed monotone vector field and prove its convergence…

Optimization and Control · Mathematics 2025-10-14 Nicholas Pischke

We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size…

Statistics Theory · Mathematics 2007-12-18 Jiming Jiang , Yihui Luan , You-Gan Wang

Iterative algorithms are fundamental tools for approximating fixed-points of nonexpansive operators in real Hilbert spaces. Among them, Krasnosel'ski\u{\i}--Mann iteration and Halpern iteration are two widely used schemes. In this work, we…

Optimization and Control · Mathematics 2026-02-20 Yifan Bai , Yantao Li , Jian Yu , Jingwei Liang

The purpose of this paper is to present an inexact version of the scaled gradient projection method on a convex set, which is inexact in two sense. First, an inexact projection on the feasible set is computed, allowing for an appropriate…

Optimization and Control · Mathematics 2021-06-10 Orizon P. Ferreira , Max V. Lemes , Leandro F. Prudente

This paper considers the inversion of ill-posed linear operators. To regularise the problem the solution is enforced to lie in a non-convex subset. Theoretical properties for the stable inversion are derived and an iterative algorithm akin…

Numerical Analysis · Mathematics 2009-11-30 Thomas Blumensath

We consider a stochastic Inverse Variational Inequality (IVI) problem defined by a continuous and co-coercive map over a closed and convex set. Motivated by the absence of performance guarantees for stochastic IVI, we present a…

Optimization and Control · Mathematics 2023-12-08 Zeinab Alizadeh , Felipe Parra Polanco , Afrooz Jalilzadeh

In this paper, we investigate the attractive properties of the proximal gradient algorithm with inertia. Notably, we show that using alternated inertia yields monotonically decreasing functional values, which contrasts with usual…

Optimization and Control · Mathematics 2018-01-18 Franck Iutzeler , Jerome Malick

We establish the weak convergence of inertial Krasnoselskii-Mann iterations towards a common fixed point of a family of quasi-nonexpansive operators, along with estimates for the non-asymptotic rate at which the residuals vanish. Strong and…

Optimization and Control · Mathematics 2023-08-23 Juan José Maulén , Ignacio Fierro , Juan Peypouquet

We propose and study a weakly convergent variant of the forward--backward algorithm for solving structured monotone inclusion problems. Our algorithm features a per-iteration deviation vector which provides additional degrees of freedom.…

Optimization and Control · Mathematics 2022-08-15 Hamed Sadeghi , Sebastian Banert , Pontus Giselsson

A type of iterative orthogonally accumulated projection methods for solving linear system of equations are proposed in this paper. This type of methods are applications of accumulated projection(AP) technique proposed recently by authors.…

Numerical Analysis · Mathematics 2016-08-24 Wujian Peng , Shuhua Zhang

A Halpern-type relaxed inertial inexact progressive hedging algorithm (PHA) is proposed for solving multi-stage stochastic variational inequalities in general probability spaces. The subproblems in this algorithm are allowed to be…

Optimization and Control · Mathematics 2024-12-10 Jiaxin Chen , Zunjie Huang , Haisen Zhang

The functional linear model extends the notion of linear regression to the case where the response and covariates are iid elements of an infinite dimensional Hilbert space. The unknown to be estimated is a Hilbert-Schmidt operator, whose…

Statistics Theory · Mathematics 2016-12-22 Tung Pham , Victor Panaretos

Convergence rates results for Tikhonov regularization of nonlinear ill-posed operator equations in abstract function spaces require the handling of both smoothness conditions imposed on the solution and structural conditions expressing the…

Numerical Analysis · Mathematics 2009-09-29 Radu Ioan Bot , Bernd Hofmann

In this paper, we propose a stochastic version of the classical Tseng's forward-backward-forward method with inertial term for solving monotone inclusions given by the sum of a maximal monotone operator and a single-valued monotone operator…

Optimization and Control · Mathematics 2022-02-22 Van Dung Nguyen , Nguyen The Vinh