General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems

Wu, Zhongming; Li, Min

doi:10.1007/s10589-019-00073-1

Cited by 42 publications

(27 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we study how to construct a general merit function D α γ (y, z, x) for step-size α and proximal parameter γ . Note that compared with the forward-backward splitting method [28,29], the DR and PR splitting methods include two implicit proximal steps, which make the DR and PR methods usually more effective when the proximal mappings are easy to evaluate [18].…”

Section: Model and Motivationmentioning

confidence: 99%

“…The assumption of KŁ functions is not a restricted one since there are huge classes of functions, for example the semi-algebraic functions, satisfy this property, see, e.g., [28,35,36]. Actually, KŁ property has been successfully and widely utilized to analyze convergence and convergence rate for many approaches such as proximalbased methods [28,29,37] and alternating minimization methods [35,36,38,39].…”

Section: Model and Motivationmentioning

confidence: 99%

See 1 more Smart Citation

Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems

2019

J Optim Theory Appl

Self Cite

View full text Add to dashboard Cite

In this paper, we propose generalized splitting methods for solving a class of nonconvex optimization problems. The new methods are extended from the classic Douglas-Rachford and Peaceman-Rachford splitting methods. The range of the new step-sizes even can be enlarged two times for some special cases. The new methods can also be used to solve convex optimization problems. In particular, for convex problems, we propose more relax conditions on step-sizes and other parameters and prove the global convergence and iteration complexity without any additional assumptions. Under the strong convexity assumption on the objective function, the linear convergence rate can be derived easily.

show abstract

Section: Model and Motivationmentioning

confidence: 99%

Section: Model and Motivationmentioning

confidence: 99%

Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems

2019

J Optim Theory Appl

Self Cite

View full text Add to dashboard Cite

show abstract

“…Therefore, they are more flexible comparing those with one inertial extrapolation while not significantly increasing the computational cost per iteration. The numerical examples in Wu and Li (2019) illustrate the computational advantage of the methods with two inertial extrapolations.…”

Section: Introductionmentioning

confidence: 99%

“…It is noteworthy that all these inertial type projection methods have a common characteristic that they only have an inertial extrapolation in each iteration. Very recently, the iterative algorithms with two inertial extrapolations were proposed, such as the general inertial proximal gradient method (Wu and Li 2019) and the general inertial Krasnosel'skiǐ-Mann iteration (Dong et al 2018a). These algorithms reduce to the methods with one inertial extrapolation by setting inertial parameters in one inertial extrapolation to be zero.…”

Section: Introductionmentioning

confidence: 99%

A general inertial projected gradient method for variational inequality problems

Dong

Liu

2021

Comp. Appl. Math.

View full text Add to dashboard Cite

The purpose of this article is to introduce a general inertial projected gradient method with a self-adaptive stepsize for solving variational inequality problems. The proposed method incorporates two different extrapolations with respect to the previous iterates into the projected gradient method. The weak convergence for our method is proved under standard assumptions without any requirement of the knowledge of the Lipschitz constant of the mapping. Furthermore, R-linear convergence rate is established under the strong monotonicity assumption of the mapping. Finally preliminary results from numerical experiments and applications to optimal control problems are performed which show the advantage of the proposed method.

show abstract

“…In [8], it has been shown that FISTA possesses O(1/k 2 ) convergence rate for the convex case which is faster than the original proximal gradient algorithm, where k counts the iteration number. For the nonconvex case, there are also some works considering the proximal gradient method with or without acceleration, see, e.g., [9][10][11][12][13][14][15]. In [13], Wen et al proved the linear convergence of proximal gradient algorithm with extrapolation for nonconvex optimization problem (1), based on the error bound condition, while the works [9-12, 14, 15] studied the proximal gradient algorithm or its variants under the Kurdyka-Łojasiewicz (KL) framework for the nonconvex case, in which they usually require some potential functions satisfying the KL property (see Definition 2.1).…”

Section: Introductionmentioning

confidence: 99%

An inexact proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth optimization problems

Jia

Dong

2019

J Inequal Appl

Self Cite

View full text Add to dashboard Cite

In this paper, we propose an inexact version of proximal gradient algorithm with extrapolation for solving a class of nonconvex nonsmooth optimization problems. Specifically, the subproblem in proximal gradient algorithm with extrapolation is allowed to be solved inexactly by certain relative error criterion, in the sense that the criterion can be updated adaptively in each iteration. Under the assumption that an auxiliary function satisfies the Kurdyka-Łojasiewicz (KL) inequality, we prove that the iterative sequence generated by the inexact proximal gradient algorithm with extrapolation converges to a stationary point of the considered problem. Furthermore, the convergence rate of the proposed algorithm can be established when the KL exponent is known. Moreover, we illustrate the advantage by applying the algorithm to solve a nonconvex optimization problem.

show abstract

General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems

Cited by 42 publications

References 41 publications

Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems

Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems

A general inertial projected gradient method for variational inequality problems

An inexact proximal gradient algorithm with extrapolation for a class of nonconvex nonsmooth optimization problems

Contact Info

Product

Resources

About