R-linear Convergence Research Articles

An inertial extragradient method for solving strongly pseudomonotone equilibrium problems in Hilbert spaces

In this work, we propose an inertial extragradient method for solving strongly pseudomonotone equilibrium problems utilizing a novel self-adaptive stepsize approach. We establish the R-linear convergence rate of the proposed method without prior knowledge of the Lipschitz-type constants associated with the bifunction. We also discuss the application of the obtained results to variational inequality problems involving strongly pseudomonotone and Lipschitz continuous mapping. Numerical examples are presented to illustrate the efficiency of the proposed method.

R-linear convergence analysis of two golden ratio projection algorithms for strongly pseudo-monotone variational inequalities

In this paper, we combine the golden ratio method with the projection algorithm to obtain the golden ratio projection algorithm. In order to get better convergence speed, we also propose an alternating golden ratio projection algorithm. Unlike ordinary inertial extrapolation, golden ratio method is constructed based on a convex combined structure about the entire iterative trajectory. The advantages of the proposed algorithms require only one projection onto the feasible set and do not require knowledge of the Lipschitz constant for the operator since our algorithms use self-adaptive step-sizes. The R-linear convergence results of the two algorithms are established for strongly pseudo-monotone variational inequality. Finally, we present some numerical experiments to show the efficiency and advantages of the proposed algorithms.

Extragradient algorithms for solving equilibrium problems on Hadamard manifolds

In this paper, we introduce three adaptive extragradient-based algorithms for solving equilibrium problems in Hadamard manifolds. The proposed algorithms can work adaptively without requiring the prior information about the Lipschitz constants of the bifunctions involved. Moreover, the iterative sequences generated by the suggested algorithms converge to the solutions of the equilibrium problems when the bifunctions are pseudomonotone and Lipschitz continuous. We also establish the global error bounds and R-linear convergence rates of the proposed algorithms in the case that the bifunctions involved are strongly pseudomonotone. Finally, a fundamental numerical example is given to illustrate the theoretical findings.

An inexact regularized proximal Newton method for nonconvex and nonsmooth optimization

This paper focuses on the minimization of a sum of a twice continuously differentiable function f and a nonsmooth convex function. An inexact regularized proximal Newton method is proposed by an approximation to the Hessian of f involving the ϱ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varrho $$\\end{document}th power of the KKT residual. For ϱ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varrho =0$$\\end{document}, we justify the global convergence of the iterate sequence for the KL objective function and its R-linear convergence rate for the KL objective function of exponent 1/2. For ϱ∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varrho \\in (0,1)$$\\end{document}, by assuming that cluster points satisfy a locally Hölderian error bound of order q on a second-order stationary point set and a local error bound of order q>1+ϱ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q>1\\!+\\!\\varrho $$\\end{document} on the common stationary point set, respectively, we establish the global convergence of the iterate sequence and its superlinear convergence rate with order depending on q and ϱ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varrho $$\\end{document}. A dual semismooth Newton augmented Lagrangian method is also developed for seeking an inexact minimizer of subproblems. Numerical comparisons with two state-of-the-art methods on ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-regularized Student’s t-regressions, group penalized Student’s t-regressions, and nonconvex image restoration confirm the efficiency of the proposed method.

The exact worst-case convergence rate of the alternating direction method of multipliers

AbstractRecently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction method of multipliers (ADMM) by using performance estimation. We give some examples which show the exactness of the given bounds. We also study the linear and R-linear convergence of ADMM in terms of dual objective. We establish that ADMM enjoys a global linear convergence rate if and only if the dual objective satisfies the Polyak–Łojasiewicz (PŁ) inequality in the presence of strong convexity. In addition, we give an explicit formula for the linear convergence rate factor. Moreover, we study the R-linear convergence of ADMM under two scenarios.

A projection method for solving monotone nonlinear equations with application

Nonlinear equations are one of the most trending research areas due to their vast applications in the sciences, social sciences, and engineering. However, the conjugate gradient method (CG) is one of the most rapidly developing iterative techniques for solving nonlinear monotone problems. Recently, much work has been done on using the CG method to solve monotone nonlinear equations. This paper discusses a new variant for solving constrained monotone nonlinear equations. The method satisfies the sufficient descent condition and proves global convergence and R-linear convergence with the help of some reasonable assumptions. In addition, two sets of numerical tests were conducted. The first experiment shows the good performance of the proposed method compared to existing methods, while the second experiment displays the performance of the proposed algorithm in compressive sensing.

Fundamental Theory and R-linear Convergence of Stretch Energy Minimization for Spherical Equiareal Parameterization

Abstract Here, we extend the finite distortion problem from bounded domains in ℝ2 to closed genus-zero surfaces in ℝ3 by a stereographic projection. Then, we derive a theoretical foundation for spherical equiareal parameterization between a simply connected closed surface M and a unit sphere 𝕊2 by minimizing the total area distortion energy on ̅ℂ. After the minimizer of the total area distortion energy is determined, it is combined with an initial conformal map to determine the equiareal map between the extended planes. From the inverse stereographic projection, we derive the equiareal map between M and 𝕊2. The total area distortion energy is rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres and is decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization function for the computation of spherical equiareal parameterization between M and 𝕊2. In addition, under relatively mild conditions, we verify that our proposed algorithm has asymptotic R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate that the assumptions for convergence always hold and indicate the efficiency, reliability, and robustness of the developed modified stretch energy minimization function.

A generalized proximal point algorithm with new step size update for solving monotone variational inequalities in real Hilbert spaces

The aim of this paper is to study a generalized version of proximal method with a new step size update for solving variational inequality problem in Hilbert spaces. Weak convergence of the proposed scheme is attained under some appropriate assumptions imposed on the operators and parameters. At the same time, we set up an R-linear convergence rate of our method when the operator is strong monotone. Numerical experiments are given to demonstrate the efficacy of our proposed iterative algorithm. Several versions of recently proposed methods in the literature are special cases of our method.

Weak and strong convergence of a modified adaptive generalized Popov's algorithm for solving variational inequality problems

In this paper, we introduce a modified adaptive step-size algorithm by basing a generalized Popov's method to solve variational inequality problems in a real Hilbert space. The algorithm only needs to act on the operator once time in each iteration, which greatly reduces the calculation time and calculation difficulty. Of particular interesting, by using our new same generalized Popov's method, we also weaken the conditions of the operator about weak continuity and quasi-monotone property. Of particular interesting, by using our new same generalized Popov's method, we can obtain a weak convergence result under the condition that the operator A is quasi-monotone and a R-linear convergence result under the condition that the operator A is strongly pseudo-monotone in the variational inequality, respectively. Finally, the feasibility and effectiveness of the algorithm are verified by some numerical experiments.

Linear convergence rate analysis of a class of exact first-order distributed methods for weight-balanced time-varying networks and uncoordinated step sizes

We analyze a class of exact distributed first order methods under a general setting on the underlying network and step-sizes. In more detail, we allow simultaneously for time-varying uncoordinated step sizes and time-varying directed weight-balanced networks, jointly connected over bounded intervals. The analyzed class of methods subsumes several existing algorithms like the unified Extra and unified DIGing (Jakovetić in IEEE Trans Signal Inf Process Netw 5(1):31–46, 2019), or the exact spectral gradient method (Jakovetić et al. in Comput Optim Appl 74:703–728, 2019) that have been analyzed before under more restrictive assumptions. Under the assumed setting, we establish R-linear convergence of the methods and present several implications that our results have on the literature. Most notably, we show that the unification strategy in Jakovetić (2019) and the spectral step-size selection strategy in Jakovetić et al. (2019) exhibit a high degree of robustness to uncoordinated time-varying step sizes and to time-varying networks.

An approximate gradient-type method for nonlinear symmetric equations with convex constraints

Based on the projection operator and the pseudo-monotone property, an approximate gradient-type method is proposed to solve nonlinear symmetric equations with convex constraints. The proposed method does not need any precise gradient or Jacobian matrix information, and only uses some approximate substitutions based on the symmetric property of equations. Under some appropriate assumptions, the global convergence and R-linear convergence rate are proved. Numerical experiments show that the proposed method is stable and effective for the test problems.

A limited memory Quasi-Newton approach for multi-objective optimization

In this paper, we deal with the class of unconstrained multi-objective optimization problems. In this setting we introduce, for the first time in the literature, a Limited Memory Quasi-Newton type method, which is well suited especially in large scale scenarios. The proposed algorithm approximates, through a suitable positive definite matrix, the convex combination of the Hessian matrices of the objectives; the update formula for the approximation matrix can be seen as an extension of the one used in the popular L-BFGS method for scalar optimization. Equipped with a Wolfe type line search, the considered method is proved to be well defined even in the nonconvex case. Furthermore, for twice continuously differentiable strongly convex problems, we state global and R-linear convergence to Pareto optimality of the sequence of generated points. The performance of the new algorithm is empirically assessed by a thorough computational comparison with state-of-the-art Newton and Quasi-Newton approaches from the multi-objective optimization literature. The results of the experiments highlight that the proposed approach is generally efficient and effective, outperforming the competitors in most settings. Moreover, the use of the limited memory method results to be beneficial within a global optimization framework for Pareto front approximation.

Dynamics based privacy preservation in decentralized optimization

With decentralized optimization having increased applications in various domains ranging from machine learning, control, to robotics, its privacy is also receiving increased attention. Existing privacy solutions for decentralized optimization achieve privacy by patching information-technology privacy mechanisms such as differential privacy or homomorphic encryption, which either sacrifices optimization accuracy or incurs heavy computation/communication overhead. We propose an inherently privacy-preserving decentralized optimization algorithm by exploiting the robustness of decentralized optimization dynamics. More specifically, we present a general decentralized optimization framework, based on which we show that the privacy of participating nodes’ gradients can be protected by adding randomness in optimization parameters. We further show that the added randomness has no influence on the accuracy of optimization, and prove that our inherently privacy-preserving algorithm has R-linear convergence when the global objective function is smooth and strongly convex. We also prove that the proposed algorithm can avoid the gradient of a node from being inferable by other nodes. Simulation results confirm the theoretical predictions.

A proximal quasi-Newton method based on memoryless modified symmetric rank-one formula

<p style='text-indent:20px;'>We consider proximal gradient methods for minimizing a composite function of a differentiable function and a convex function. To accelerate the general proximal gradient methods, we focus on proximal quasi-Newton type methods based on proximal mappings scaled by quasi-Newton matrices. Although it is usually difficult to compute the scaled proximal mappings, applying the memoryless symmetric rank-one (SR1) formula makes this easier. Since the scaled (quasi-Newton) matrices must be positive definite, we develop an algorithm using the memoryless SR1 formula based on a modified spectral scaling secant condition. We give the subsequential convergence property of the proposed method for general objective functions. In addition, we show the R-linear convergence property of the method under a strong convexity assumption. Finally, some numerical results are reported.</p>

SIGNAL RECOVERY WITH CONSTRAINED MONOTONE NONLINEAR EQUATIONS THROUGH AN EFFECTIVE THREE-TERM CONJUGATE GRADIENT METHOD

In this paper, we introduce a three-term conjugate gradient-type projection method for solving constrained monotone nonlinear equations. In this method, we firstly undertake the transformation of the relative matrices proposed by Yao and Ning. Secondly, we obtain the new relative matrices involving two parameters. Subsequently, we construct a relationship for the two parameters via the quasi-Newton equation and obtain the parameters by simplifying maximum eigenvalue of the new relative matrices. Finally, combining the three-term conjugate gradient method with projection technique, we establish an efficient three-term conjugate gradient-type projection algorithm. Meanwhile, we also give some theoretical analysis about the global convergence and R-linear convergence of the proposed algorithm under quite reasonable technical assumptions. Performance comparisons show that our proposed method is competitive and efficient for solving large-scale nonlinear monotone equations with convex constraints. Furthermore, the presented algorithm is also applied to recovery of a sparse signal in compressive sensing, and obtain practical, efficient and competitive performance in comparing with state-of-the-art algorithms.

New extrapolation projection contraction algorithms based on the golden ratio for pseudo-monotone variational inequalities

<abstract><p>In real Hilbert spaces, for the purpose of trying to deal with the pseudo-monotone variational inequalities problem, we present a new extrapolation projection contraction algorithm based on the golden ratio in this study. Unlike ordinary inertial extrapolation, the algorithms are constructed based on a convex combined structure about the entire iterative trajectory. Extrapolation parameter $ \psi $ is selected in a more relaxed range instead of only taking the golden ratio $ \phi = \frac{\sqrt{5}+1 }{2} $ as the upper bound. Second, we propose an alternating extrapolation projection contraction algorithm to better increase the convergence effects of the extrapolation projection contraction algorithm based on the golden ratio. All our algorithms employ non-constantly decreasing adaptive step-sizes. The weak convergence results of the two algorithms are established for the pseudo-monotone variational inequalities. Additionally, the R-linear convergence results are investigated for strongly pseudo-monotone variational inequalities. Finally, we show the validity and superiority of the suggested methods with several numerical experiments. The numerical results show that alternating extrapolation does have obvious acceleration effect in practical application compared with no alternating extrapolation. Thus, the obvious effect of relaxing the selection range of parameter $ \psi $ on our two algorithms is clearly demonstrated.</p></abstract>

Convergence Rates for the Relaxed Peaceman-Rachford Splitting Method on a Monotone Inclusion Problem

We consider the convergence behavior using the relaxed Peaceman–Rachford splitting method to solve the monotone inclusion problem 0 in (A + B)(u), where A, B: Re ^n rightrightarrows Re ^n are maximal beta -strongly monotone operators, n ge 1 and beta > 0. Under a technical assumption, convergence of iterates using the method on the problem is proved when either A or B is single-valued, and the fixed relaxation parameter theta lies in the interval (2 + beta , 2 + beta + min { beta , 1/beta }). With this convergence result, we address an open problem that is not settled in Monteiro et al. (Computat Optim Appl 70:763–790, 2018) on the convergence of these iterates for theta in (2 + beta , 2 + beta + min { beta , 1/beta }). Pointwise convergence rate results and R-linear convergence rate results when theta lies in the interval [2 + beta , 2 + beta + min {beta , 1/beta }) are also provided in the paper. Our analysis to achieve these results is atypical and hence novel. Numerical experiments on the weighted Lasso minimization problem are conducted to test the validity of the assumption.

Fast relaxed inertial Tseng’s method-based algorithm for solving variational inequality and fixed point problems in Hilbert spaces

Motivated and inspired by the works of Ceng et al. (2010) and Yao and Postolache (2012), we first study a relaxed inertial Tseng’s method for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of fixed points of an κ-demicontractive mapping in real Hilbert spaces. The strong convergence of the algorithm is proved with conditions weaker than the conditions of other methods studied in the literature. Next, we also obtain an R-linear convergence rate of relaxed inertial Tseng’s method under strong pseudomonotonicity and Lipschitz continuity assumptions of the variational inequality mapping. As far as we know, these results have not been considered before in the literature. Finally, some numerical examples illustrate the effectiveness of our algorithms.

Reward-Weighted Regression Converges to a Global Optimum

Reward-Weighted Regression (RWR) belongs to a family of widely known iterative Reinforcement Learning algorithms based on the Expectation-Maximization framework. In this family, learning at each iteration consists of sampling a batch of trajectories using the current policy and fitting a new policy to maximize a return-weighted log-likelihood of actions. Although RWR is known to yield monotonic improvement of the policy under certain circumstances, whether and under which conditions RWR converges to the optimal policy have remained open questions. In this paper, we provide for the first time a proof that RWR converges to a global optimum when no function approximation is used, in a general compact setting. Furthermore, for the simpler case with finite state and action spaces we prove R-linear convergence of the state-value function to the optimum.

Two fast converging inertial subgradient extragradient algorithms with variable stepsizes for solving pseudo-monotone VIPs in Hilbert spaces

In this work, we propose two new iterative schemes for finding an element of the set of solutions of a pseudo-monotone, Lipschitz continuous variational inequality problem in real Hilbert spaces. The weak and strong convergence theorems are presented. The advantage of the proposed algorithms is that they do not require prior knowledge of the Lipschitz constant of the variational inequality mapping and only compute one projection onto a feasible set per iteration as well as without using the sequentially weakly continuity of the associated mapping. Under additional strong pseudo-monotonicity and Lipschitz continuity assumptions, we obtain also an R-linear convergence rate of the proposed algorithm. Finally, some numerical examples are given to illustrate the effectiveness of the algorithms.