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The proximal gradient method is a standard approach for solving composite minimization problems in which the objective function is the sum of a continuously differentiable function and a lower semicontinuous, extended-valued function. The…

Optimization and Control · Mathematics 2025-05-02 Xiaoxi Jia , Kai Wang

We consider a variable metric linesearch based proximal gradient method for the minimization of the sum of a smooth, possibly nonconvex function plus a convex, possibly nonsmooth term. We prove convergence of this iterative algorithm to a…

Numerical Analysis · Mathematics 2017-04-11 Silvia Bonettini , Ignace Loris , Federica Porta , Marco Prato , Simone Rebegoldi

We consider a composite optimization problem where the sum of a continuously differentiable and a merely lower semicontinuous function has to be minimized. The proximal gradient algorithm is the classical method for solving such a problem…

Optimization and Control · Mathematics 2023-05-01 Xiaoxi Jia , Christian Kanzow , Patrick Mehlitz

This work investigates a Bregman and inertial extension of the forward-reflected-backward algorithm [Y. Malitsky and M. Tam, SIAM J. Optim., 30 (2020), pp. 1451--1472] applied to structured nonconvex minimization problems under relative…

Optimization and Control · Mathematics 2024-04-17 Ziyuan Wang , Andreas Themelis , Hongjia Ou , Xianfu Wang

Nonconvex and nonsmooth optimization problems are important and challenging for statistics and machine learning. In this paper, we propose Projected Proximal Gradient Descent (PPGD) which solves a class of nonconvex and nonsmooth…

Optimization and Control · Mathematics 2024-09-26 Yingzhen Yang , Ping Li

In this work, we develop a level-set subdifferential error bound condition aiming towards convergence rate analysis of a variable Bregman proximal gradient (VBPG) method for a broad class of nonsmooth and nonconvex optimization problems. It…

Optimization and Control · Mathematics 2020-09-01 Daoli Zhu , Sien Deng , Minghua Li , Lei Zhao

In this paper, we consider a class of generalized difference-of-convex functions (DC) programming, whose objective is the difference of two convex (not necessarily smooth) functions plus a decomposable (possibly nonconvex) function with…

Optimization and Control · Mathematics 2024-09-10 Chenjian Pan , Yingxin Zhou , Hongjin He , Chen Ling

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…

Optimization and Control · Mathematics 2013-08-28 Ting Kei Pong

In order to solve the minimization of a nonsmooth convex function, we design an inertial second-order dynamic algorithm, which is obtained by approximating the nonsmooth function by a class of smooth functions. By studying the asymptotic…

Optimization and Control · Mathematics 2021-12-20 Xin Qu , Wei Bian

Matrix Factorization is a popular non-convex optimization problem, for which alternating minimization schemes are mostly used. They usually suffer from the major drawback that the solution is biased towards one of the optimization…

Optimization and Control · Mathematics 2019-12-09 Mahesh Chandra Mukkamala , Peter Ochs

In this paper, we consider a class of difference-of-convex (DC) optimization problems, which require only a weaker restricted $L$-smooth adaptable property on the smooth part of the objective function, instead of the standard global…

Optimization and Control · Mathematics 2025-04-30 Lei Yang , Jingjing Hu , Kim-Chuan Toh

An algorithm is proposed, analyzed, and tested for minimizing locally Lipschitz objective functions that may be nonconvex and/or nonsmooth. The algorithm, which is built upon the gradient-sampling methodology, is designed specifically for…

Optimization and Control · Mathematics 2026-04-02 Albert S. Berahas , Frank E. Curtis , Lara Zebiane

We study the problem of minimizing a relatively-smooth convex function using stochastic Bregman gradient methods. We first prove the convergence of Bregman Stochastic Gradient Descent (BSGD) to a region that depends on the noise (magnitude…

Optimization and Control · Mathematics 2021-04-21 Radu-Alexandru Dragomir , Mathieu Even , Hadrien Hendrikx

We focus on the problem of minimizing the sum of smooth component functions (where the sum is strongly convex) and a non-smooth convex function, which arises in regularized empirical risk minimization in machine learning and distributed…

Optimization and Control · Mathematics 2016-08-08 Nuri Denizcan Vanli , Mert Gurbuzbalaban , Asu Ozdaglar

We consider the composite minimization problem with the objective function being the sum of a continuously differentiable and a merely lower semicontinuous and extended-valued function. The proximal gradient method is probably the most…

Optimization and Control · Mathematics 2024-11-20 Christian Kanzow , Leo Lehmann

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

We propose a Bregman inertial forward-reflected-backward (BiFRB) method for nonconvex composite problems. Our analysis relies on a novel approach that imposes general conditions on implicit merit function parameters, which yields a stepsize…

Optimization and Control · Mathematics 2022-07-05 Xianfu Wang , Ziyuan Wang

This paper focuses on the problem of minimizing a locally Lipschitz continuous function. Motivated by the effectiveness of Bregman gradient methods in training nonsmooth deep neural networks and the recent progress in stochastic subgradient…

Optimization and Control · Mathematics 2025-06-02 Kuangyu Ding , Kim-Chuan Toh

We investigate two inertial forward-backward algorithms in connection with the minimization of the sum of a non-smooth and possibly non-convex and a non-convex differentiable function. The algorithms are formulated in the spirit of the…

Functional Analysis · Mathematics 2021-01-20 Szilárd Csaba László

This paper proposes and develops inexact proximal methods for finding stationary points of the sum of a smooth function and a nonsmooth weakly convex one, where an error is present in the calculation of the proximal mapping of the nonsmooth…

Optimization and Control · Mathematics 2023-08-07 Pham Duy Khanh , Boris Mordukhovich , Vo Thanh Phat , Dat Ba Tran