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AN INDEFINITE-PROXIMAL-BASED STRICTLY CONTRACTIVE PEACEMAN-RACHFORD SPLITTING METHOD

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Journal of Computational Mathematics 2023年第6期41卷 1017-1040页

作者： Yan Gu Bo Jiang deren han Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing210016China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing210016China School of Mathematical Sciences Key Laboratory for NSLSCS of Jiangsu ProvinceNanjing Normal UniversityNanjing210023China LMIB School of Mathematics and Systems ScienceBeihang UniversityBeijing100191China Key Laboratory for NSLSCS of Jiangsu Province China

The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear ***,its convergence was not guaranteed without extra *** et al.(SIAM ***.24:1011-1040,2014)proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation *** this paper,we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation ***,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 *** 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 *** numerical tests on the l 1 regularized least square problem demonstrate the efficiency of the proposed method.

关键词： Indefinite proximal Strictly contractive Peaceman-Rachford splitting method Convex minimization Convergence rate

O(1/t) complexity analysis of the generalized alternating direction method of multipliers

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Science China Mathematics 2019年第4期62卷 795-808页

作者： Xingju Cai deren han School of Mathematical Sciences and Key Laboratory for Numerical Simulation of Large Scale Complex Systems of Jiangsu Province Nanjing Normal University

Owing to its efficiency in solving some types of large-scale separable optimization problems with linear constraints, the convergence rate of the alternating direction method of multipliers(ADMM for short) has recently attracted significant attention. In this paper, we consider the generalized ADMM(G-ADMM), which incorporates an acceleration factor and is more efficient. Instead of using a solution measure that depends on a bounded set and cannot be easily estimated, we propose using the original ?-optimal solution measure, under which we prove that the G-ADMM converges at a rate of O(1/t). The new bound depends on the penalty parameter and the distance between the initial point and the solution set, which is more reasonable than the previous bound.

关键词： generalized alternating direction method of multipliers separable convex optimization iteration complexity sublinear convergence rate

A successive approximation method for quantum separability

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Frontiers of Mathematics in China 2013年第6期8卷 1275-1293页

作者： deren han Liqun QI School of Mathematical Sciences Jiangsu Key Laboratory for NSLSCS Nanjing Normal University Nanjing 210023 China Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong China

Determining whether a quantum state is separable or inseparable （entangled） is a problem of fundamental importance in quantum science and has attracted much attention since its first recognition by Einstein, Podolsky and Rosen [Phys. Rev., 1935, 47： 777] and SchrSdinger [Naturwissenschaften, 1935, 23： 807-812, 823-828, 844-849]. In this paper, we propose a successive approximation method （SAM） for this problem, which approximates a given quantum state by a so-called separable state： if the given states is separable, this method finds its rank-one components and the associated weights; otherwise, this method finds the distance between the given state to the set of separable states, which gives information about the degree of entanglement in the system. The key task per iteration is to find a feasible descent direction, which is equivalent to finding the largest M-eigenvalue of a fourth-order tensor. We give a direct method for this problem when the dimension of the tensor is 2 and a heuristic cross-hill method for cases of high dimension. Some numerical results and experiences are presented.

关键词： Quantum system entanglement tensor successive approximation,M-eigenvalue cross-hill

Prediction-correction method with BB step sizes

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Frontiers of Mathematics in China 2018年第6期13卷 1325-1340页

作者： Xiaomei DONG Xingju CAI deren han Jiangsu Key Laboratory for NSLSCS School of Mathematical Sciences Nanjing Normal UniversityNanjing 210023China School of Mathematics and Systems Science Beihang UniversityBeijing 100191China

In the prediction-correction method for variational inequality (VI) problems, the step size selection plays an important role for its performance. In this paper, we employ the Barzilai-Borwein (BB) strategy in the prediction step, which is efficient for many optimization problems from a computational point of view. To guarantee the convergence, we adopt the line search technique, and relax the conditions to accept the BB step sizes as large as possible. In the correction step, we utilize a longer step length to calculate the next iteration point. Finally, we present some preliminary numerical results to show the efficiency of the algorithms.

关键词： BB step sizes projection method prediction-correction method line search 65K05 90C30 90C33

A TRUST-REGION-BASED ALTERNATING LEAST-SQUARES ALGORITHM FOR TENSOR DECOMPOSITIONS

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Journal of Computational Mathematics 2018年第3期36卷 351-373页

作者： Fan Jiang deren han Xiaofei Zhang Jiangsu Key Laboratory of NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing 210023 China Information Technology Department Chinascope Nanjing 210023 China

Tensor canonical decomposition （shorted as CANDECOMP/PARAFAC or CP） decomposes a tensor as a sum of rank-one tensors, which finds numerous applications in signal processing, hypergraph analysis, data analysis, etc. Alternating least-squares （ALS） is one of the most popular numerical algorithms for solving it. While there have been lots of efforts for enhancing its efficiency, in general its convergence can not been guaranteed. In this paper, we cooperate the ALS and the trust-region technique from optimization field to generate a trust-region-based alternating least-squares （TRALS） method for CP. Under mild assumptions, we prove that the whole iterative sequence generated by TRALS converges to a stationary point of CP. This thus provides a reasonable way to alleviate the swamps, the notorious phenomena of ALS that slow down the speed of the algorithm. Moreover, the trust region itself, in contrast to the regularization alternating least-squares （RALS） method, provides a self-adaptive way in choosing the parameter, which is essential for the efficiency of the algorithm. Our theoretical result is thus stronger than that of RALS in [26], which only proved the cluster point of the iterative sequence generated by RALS is a stationary point. In order to accelerate the new algorithm, we adopt an extrapolation scheme. We apply our algorithm to the amino acid fluorescence data decomposition from chemometrics, BCM decomposition and rank-（Lr, Lr, 1） decomposition arising from signal processing, and compare it with ALS and RALS. The numerical results show that TRALS is superior to ALS and RALS, both from the number of iterations and CPU time perspectives.

关键词： Tensor decompositions Trust region method Alternating least-squares Ex-trapolation scheme Global convergence Regularization.

Nonnegative tensor factorizations using an alternating direction method

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Frontiers of Mathematics in China 2013年第1期8卷 3-18页

作者： Xingju CAI Yannan CHEN deren han School of Mathematical Sciences Key Laboratory for NSLSCS of Jiangsu ProvinceNanjing Normal University Nanjing 210023 China College of Science Nanjing Forestry University Nanjing 210037 China

The nonnegative tensor （matrix） factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method.

关键词： Nonnegative matrix factorization nonnegative tensor factorization,nonnegative least squares alternating direction method

Convergence of ADMM for multi-block nonconvex separable optimization models

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Frontiers of Mathematics in China 2017年第5期12卷 1139-1162页

作者： Ke GUO deren han David Z. W. WANG Tingting WU School of Mathematical Sciences and Key Laboratory for NSLSCS of Jiangsu Province Nanjing Normal University Nanjing 210023 China School of Civil and Environmental Engineering Nanyang Technological UniversitySingapore 639798 Singapore School of Science Nanjing University of Posts and Telecommunications Nanjing 210023China

For solving minimization problems whose objective function is the sum of two functions without coupled variables and the constrained function is linear, the alternating direction method of multipliers （ADMM） has exhibited its efficiency and its convergence is well understood. When either the involved number of separable functions is more than two, or there is a nonconvex function~ ADMM or its direct extended version may not converge. In this paper, we consider the multi-block *** optimization problems with linear constraints and absence of convexity of the involved component functions. Under the assumption that the associated function satisfies the Kurdyka- Lojasiewicz inequality, we prove that any cluster point of the iterative sequence generated by ADMM is a critical point, under the mild condition that the penalty parameter is sufficiently large. We also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm.

关键词： Nonconvex optimization separable structure alternating directionmethod of rnultip!iers (.ADMM) Kurdyka-Lojasiewicz inequality

求解二阶锥绝对值方程组的非单调光滑牛顿算法

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计算数学 2023年第2期45卷 251-266页

作者：于冬梅王增伟陈彩荣韩德仁辽宁工程技术大学理学院阜新123000 福建师范大学数学与统计学院福州350117 北京航空航天大学数学科学学院信息与行为教育部重点实验室北京100191

本文提出了求解二阶锥绝对值方程组(SOCAVE)的非单调光滑牛顿算法.在适当的条件下分析了算法的全局收敛性和局部二次收敛性.数值结果表明用非单调光滑牛顿算法求解SOCAVE是可行且高效的.

关键词：绝对值方程组二阶锥光滑牛顿算法非单调线搜索

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多模式交通均衡问题的一阶分裂算法

运筹学学报 2023年第2期27卷 63-78页

作者：王茂然蔡邢菊吴中明韩德仁南京师范大学数学科学学院江苏南京210023 南京信息工程大学管理工程学院江苏南京210044 北京航空航天大学数学科学学院北京100191

本文研究包含私人交通和公共交通工具的多模式交通均衡问题,将其建模成带线性不等式约束的可分单调变分不等式问题,并提出一种修正的交替方向乘子法进行求解。通过适当地修改子问题并加上一个简单的校正步,提出一种针对线性不等式约束... 详细信息

本文研究包含私人交通和公共交通工具的多模式交通均衡问题,将其建模成带线性不等式约束的可分单调变分不等式问题,并提出一种修正的交替方向乘子法进行求解。通过适当地修改子问题并加上一个简单的校正步,提出一种针对线性不等式约束问题的并行求解算法。在一般的假设条件下,证明了这个新算法的全局收敛性和次线性收敛速度,并把算法应用到交通模型中。

关键词：交通均衡问题变分不等式交替方向乘子法全局收敛次线性收敛