arXiv subjects

Deren Han

Publications and source records attributed to Deren Han.

52 records · Page 3Linked to original sources

arXiv2023

On the convergence analysis of the greedy randomized Kaczmarz method

In this paper, we analyze the greedy randomized Kaczmarz (GRK) method proposed in Bai and Wu (SIAM J. Sci. Comput., 40(1):A592--A606, 2018) for solving linear systems. We develop more precise greedy probability criteria to effectively select the working row from the coefficient matrix. Notably, we prove that the linear convergence of the GRK method is deterministic and demonstrate that using a tighter threshold parameter can lead to a faster convergence rate. Our result revises existing convergence analyses, which are solely based on the expected error by realizing that the iterates of the GRK method are random variables. Consequently, we obtain an improved iteration complexity for the GRK method. Moreover, the Polyak's heavy ball momentum technique is incorporated to improve the performance of the GRK method. We propose a refined convergence analysis, compared with the technique used in Loizou and Richtárik (Comput. Optim. Appl., 77(3):653--710, 2020), of momentum variants of randomized iterative methods, which shows that the proposed GRK method with momentum (mGRK) also enjoys a deterministic linear convergence. Numerical experiments show that the mGRK method is more efficient than the GRK method.

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arXiv2023

The neural network models with delays for solving absolute value equations

An inverse-free neural network model with mixed delays is proposed for solving the absolute value equation (AVE) $Ax -|x| - b =0$, which includes an inverse-free neural network model with discrete delay as a special case. By using the Lyapunov-Krasovskii theory and the linear matrix inequality (LMI) method, the developed neural network models are proved to be exponentially convergent to the solution of the AVE. Compared with the existing neural network models for solving the AVE, the proposed models feature the ability of solving a class of AVE with $\|A^{-1}\|>1$. Numerical simulations are given to show the effectiveness of the two delayed neural network models.

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arXiv2023

Randomized Kaczmarz method with adaptive stepsizes for inconsistent linear systems

We investigate the randomized Kaczmarz method that adaptively updates the stepsize using readily available information for solving inconsistent linear systems. A novel geometric interpretation is provided which shows that the proposed method can be viewed as an orthogonal projection method in some sense. We prove that this method converges linearly in expectation to the unique minimum Euclidean norm least-squares solution of the linear system, and provide a tight upper bound for the convergence of the proposed method. Numerical experiments are also given to illustrate the theoretical results.

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arXiv2023

Improving the generalization via coupled tensor norm regularization

In this paper, we propose a coupled tensor norm regularization that could enable the model output feature and the data input to lie in a low-dimensional manifold, which helps us to reduce overfitting. We show this regularization term is convex, differentiable, and gradient Lipschitz continuous for logistic regression, while nonconvex and nonsmooth for deep neural networks. We further analyze the convergence of the first-order method for solving this model. The numerical experiments demonstrate that our method is efficient.

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arXiv2023

Understanding the convergence of the preconditioned PDHG method: a view of indefinite proximal ADMM

The primal-dual hybrid gradient (PDHG) algorithm is popular in solving min-max problems which are being widely used in a variety of areas. To improve the applicability and efficiency of PDHG for different application scenarios, we focus on the preconditioned PDHG (PrePDHG) algorithm, which is a framework covering PDHG, alternating direction method of multipliers (ADMM), and other methods. We give the optimal convergence condition of PrePDHG in the sense that the key parameters in the condition can not be further improved, which fills the theoretical gap in the-state-of-art convergence results of PrePDHG, and obtain the ergodic and non-ergodic sublinear convergence rates of PrePDHG. The theoretical analysis is achieved by establishing the equivalence between PrePDHG and indefinite proximal ADMM. Besides, we discuss various choices of the proximal matrices in PrePDHG and derive some interesting results. For example, the convergence condition of diagonal PrePDHG is improved to be tight, the dual stepsize of the balanced augmented Lagrangian method can be enlarged to $4/3$ from $1$, and a balanced augmented Lagrangian method with symmetric Gauss-Seidel iterations is also explored. Numerical results on the matrix game, projection onto the Birkhoff polytope, earth mover's distance, and CT reconstruction verify the effectiveness and superiority of PrePDHG.

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arXiv2022

An indefinite-proximal-based strictly contractive Peaceman-Rachford splitting method

The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints. However, its convergence was not guaranteed without extra requirements. He {\it et al.} (SIAM J. Optim. 24: 1011 - 1040, 2014) proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor. In this paper, we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors. Besides, motivated by the recent advances on the ADMM type method with indefinite proximal terms, we employ the indefinite proximal term in the strictly contractive Peaceman-Rachford splitting method. We show that the proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is convergent and also prove the $o(1/t)$ convergence rate in the nonergodic sense. The numerical tests on the $l_1$ regularized least square problem demonstrate the efficiency of the proposed method.

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arXiv2022

On pseudoinverse-free randomized methods for linear systems: Unified framework and acceleration

We present a new framework for the analysis and design of randomized algorithms for solving various types of linear systems, including consistent or inconsistent, full rank or rank-deficient. Our method is formulated with four randomized sampling parameters, which allows the method to cover many existing randomization algorithms within a unified framework, including the doubly stochastic Gauss-Seidel, randomized Kaczmarz method, randomized coordinate descent method, and Gaussian Kaczmarz method. Compared with the projection-based block algorithms where a pseudoinverse for solving a least-squares problem is utilized at each iteration, our design is pseudoinverse-free. Furthermore, the flexibility of the new approach also enables the design of a number of new methods as special cases. Polyak's heavy ball momentum technique is also introduced in our framework for improving the convergence behavior of the method. We prove the global linear convergence rates of our method as well as an accelerated linear rate for the case of the norm of expected iterates. Finally, numerical experiments are provided to confirm our results.

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arXiv2022

A fixed-time inverse-free dynamical system for solving the system of absolute value equations

In this paper, an inverse-free dynamical system with fixed-time convergence is presented to solve the system of absolute value equations (AVEs). Under a mild condition, it is proved that the solution of the proposed dynamical system converges to the solution of the AVEs. Moreover, in contrast to the existing inverse-free dynamical system \cite{chen2021}, a conservative settling-time of the proposed method is given. Numerical simulations illustrate the effectiveness of the new method.

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arXiv2022

A dynamical system based on projection operator for solving absolute value equations associated with second-order cone

A new equivalent reformulation of the absolute value equations associated with second-order cone (SOCAVEs) is emphasised, from which a dynamical system based on projection operator for solving SOCAVEs is constructed. Under proper assumptions, the equilibrium points of the dynamical system exist and could be (globally) asymptotically stable. Some numerical simulations are given to show the effectiveness of the proposed method.

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arXiv2022

On finite termination of the generalized Newton method for solving absolute value equations

Motivated by the framework constructed by Brugnano and Casulli $[$SIAM J. Sci. Comput. 30: 463--472, 2008$]$, we analyze the finite termination property of the generalized Netwon method (GNM) for solving the absolute value equation (AVE). More precisely, for some special matrices, GNM is terminated in at most $2n + 2$ iterations. A new result for the unique solvability and unsolvability of the AVE is obtained. Numerical experiments are given to demonstrate the theoretical analysis.

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arXiv2022

An inexact framework of the Newton-based matrix splitting iterative method for the generalized absolute value equation

An inexact framework of the Newton-based matrix splitting (INMS) iterative method is developed to solve the generalized absolute value equation, whose exact version was proposed by Zhou, Wu and Li [H.-Y. Zhou, S.-L. Wu and C.-X. Li, \textit{J. Comput. Appl. Math.}, 394 (2021), 113578]. Global linear convergence of the INMS iterative method is investigated in detail. Some numerical results are given to show the superiority of the INMS iterative method.

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arXiv2021

Convergence on a symmetric accelerated stochastic ADMM with larger stepsizes

In this paper, we develop a symmetric accelerated stochastic Alternating Direction Method of Multipliers (SAS-ADMM) for solving separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and an average function of many smooth convex functions. Our proposed algorithm combines both ideas of ADMM and the techniques of accelerated stochastic gradient methods possibly with variance reduction to solve the smooth subproblem. One main feature of SAS-ADMM is that its dual variable is symmetrically updated after each update of the separated primal variable, which would allow a more flexible and larger convergence region of the dual variable compared with that of standard deter-ministic or stochastic ADMM. This new stochastic optimization algorithm is shown to have ergodic converge in expectation with O(1/T) convergence rate, where T is the number of outer iterations. Our preliminary experiments indicate the proposed algorithm is very effective for solving separable optimization problems from big-data applications. Finally, 3-block extensions of the algorithm and its variant of an accelerated stochastic augmented Lagrangian method are discussed in the appendix.

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arXiv2021

A non-monotone smoothing Newton algorithm for solving the system of generalized absolute value equations

The system of generalized absolute value equations (GAVE) has attracted more and more attention in the optimization community. In this paper, by introducing a smoothing function, we develop a smoothing Newton algorithm with non-monotone line search to solve the GAVE. We show that the non-monotone algorithm is globally and locally quadratically convergent under a weaker assumption than those given in most existing algorithms for solving the GAVE. Numerical results are given to demonstrate the viability and efficiency of the approach.

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arXiv2021

An inexact Douglas-Rachford splitting method for solving absolute value equations

The last two decades witnessed the increasing of the interests on the absolute value equations (AVE) of finding $x\in\mathbb{R}^n$ such that $Ax-|x|-b=0$, where $A\in \mathbb{R}^{n\times n}$ and $b\in \mathbb{R}^n$. In this paper, we pay our attention on designing efficient algorithms. To this end, we reformulate AVE to a generalized linear complementarity problem (GLCP), which, among the equivalent forms, is the most economical one in the sense that it does not increase the dimension of the variables. For solving the GLCP, we propose an inexact Douglas-Rachford splitting method which can adopt a relative error tolerance. As a consequence, in the inner iteration processes, we can employ the LSQR method ([C.C. Paige and M.A. Saunders, ACM Trans. Mathe. Softw. (TOMS), 8 (1982), pp. 43--71]) to find a qualified approximate solution for each subproblem, which makes the cost per iteration very low. We prove the convergence of the algorithm and establish its global linear rate of convergence. Comparing results with the popular algorithms such as the exact generalized Newton method [O.L. Mangasarian, Optim. Lett., 1 (2007), pp. 3--8], the inexact semi-smooth Newton method [J.Y.B. Cruz, O.P. Ferreira and L.F. Prudente, Comput. Optim. Appl., 65 (2016), pp. 93--108] and the exact SOR-like method [Y.-F. Ke and C.-F. Ma, Appl. Math. Comput., 311 (2017), pp. 195--202] are reported, which indicate that the proposed algorithm is very promising. Moreover, our method also extends the range of numerically solvable of the AVE; that is, it can deal with not only the case that $\|A^{-1}\|<1$, the commonly used in those existing literature, but also the case where $\|A^{-1}\|=1$.

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arXiv2021

Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations

The SOR-like iteration method for solving the absolute value equations~(AVE) of finding a vector $x$ such that $Ax - |x| - b = 0$ with $ν= \|A^{-1}\|_2 < 1$ is investigated. The convergence conditions of the SOR-like iteration method proposed by Ke and Ma ([{\em Appl. Math. Comput.}, 311:195--202, 2017]) are revisited and a new proof is given, which exhibits some insights in determining the convergent region and the optimal iteration parameter. Along this line, the optimal parameter which minimizes $\|T_ν(ω)\|_2$ with $$T_ν(ω) = \left(\begin{array}{cc} |1-ω| & ω^2ν\\ |1-ω| & |1-ω| +ω^2ν\end{array}\right)$$ and the approximate optimal parameter which minimizes $η_ν(ω) =\max\{|1-ω|,νω^2\}$ are explored. The optimal and approximate optimal parameters are iteration-independent and the bigger value of $ν$ is, the smaller convergent region of the iteration parameter $ω$ is. Numerical results are presented to demonstrate that the SOR-like iteration method with the optimal parameter is superior to that with the approximate optimal parameter proposed by Guo, Wu and Li ([{\em Appl. Math. Lett.}, 97:107--113, 2019]). In some situation, the SOR-like itration method with the optimal parameter performs better, in terms of CPU time, than the generalized Newton method (Mangasarian, [{\em Optim. Lett.}, 3:101--108, 2009]) for solving the AVE.

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arXiv2015

Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Quadratic and Semi-Definite Programming

In this paper, we aim to provide a comprehensive analysis on the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a certain error bound condition, we establish the global linear rate of convergence for a more general semi-proximal ADMM with the dual steplength being restricted to be in the open interval $(0, (1+\sqrt{5})/2)$. In our analysis, we assume neither the strong convexity nor the strict complementarity except an error bound condition, which holds automatically for convex composite quadratic programming. This semi-proximal ADMM, which includes the classic ADMM, not only has the advantage to resolve the potentially non-solvability issue of the subproblems in the classic ADMM but also possesses the abilities of handling multi-block convex optimization problems efficiently. We shall use convex composite quadratic programming and quadratic semi-definite programming as important applications to demonstrate the significance of the obtained results. Of its own novelty in second-order variational analysis, a complete characterization is provided on the isolated calmness for the nonlinear convex semi-definite optimization problem in terms of its second order sufficient optimality condition and the strict Robinson constraint qualification for the purpose of proving the linear rate convergence of the semi-proximal ADMM when applied to two- and multi-block convex quadratic semi-definite programming.

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