This paper proposes a semismooth Newton method for a class of bilinear programming problems(BLPs)based on the augmented Lagrangian,in which the BLPs are reformulated as a system of nonlinear equations with original va...This paper proposes a semismooth Newton method for a class of bilinear programming problems(BLPs)based on the augmented Lagrangian,in which the BLPs are reformulated as a system of nonlinear equations with original variables and Lagrange multipliers.Without strict complementarity,the convergence of the method is studied by means of theories of semismooth analysis under the linear independence constraint qualification and strong second order sufficient condition.At last,numerical results are reported to show the performance of the proposed method.展开更多
Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the sys...Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the system of nonsmooth equations, we develop a semismooth Newton method for the tensor complementarity problem. We prove the monotone convergence theorem for the proposed method under proper conditions.展开更多
Based on the generalized Fischer-Burmeister function, Chen et al in 2008 put forward a regularization semismooth Newton method for solving the nonlinear complementarity problem with a P0-function. In this paper, we in...Based on the generalized Fischer-Burmeister function, Chen et al in 2008 put forward a regularization semismooth Newton method for solving the nonlinear complementarity problem with a P0-function. In this paper, we investigate the above algorithm with the monotone line search replaced by a non-monotone line search. It is shown that the non-monotone algorithm is well-defined, and is globally and locally superlinearly convergent under standard assumptions.展开更多
This paper develops and analyzes multigrid semismooth Newton methods for a class of inequality-constrained optimization problems in function space which are motivated by and include linear elastic contact problems of ...This paper develops and analyzes multigrid semismooth Newton methods for a class of inequality-constrained optimization problems in function space which are motivated by and include linear elastic contact problems of Signorini type. We show that after a suitable Moreau-Yosida type regularization of the problem superlinear local convergence is obtained for a class of semismooth Newton methods. In addition, estimates for the order of tile error introduced by the regularization are derived. The main part of the paper is devoted to the analysis of a multilevel preconditioner for the semismooth Newton system. We prove a rigorous bound for the contraction rate of the multigrid cycle which is robust with respect to sufficiently small regularization parameters and the number of grid levels. Moreover, it applies to adaptively refined grids. The paper concludes with numerical results.展开更多
In this work,we present probabilistic local convergence results for a stochastic semismooth Newton method for a class of stochastic composite optimization problems involving the sum of smooth nonconvex and nonsmooth c...In this work,we present probabilistic local convergence results for a stochastic semismooth Newton method for a class of stochastic composite optimization problems involving the sum of smooth nonconvex and nonsmooth convex terms in the objective function.We assume that the gradient and Hessian information of the smooth part of the objective function can only be approximated and accessed via calling stochastic firstand second-order oracles.The approach combines stochastic semismooth Newton steps,stochastic proximal gradient steps and a globalization strategy based on growth conditions.We present tail bounds and matrix concentration inequalities for the stochastic oracles that can be utilized to control the approximation errors via appropriately adjusting or increasing the sampling rates.Under standard local assumptions,we prove that the proposed algorithm locally turns into a pure stochastic semismooth Newton method and converges r-linearly or r-superlinearly with high probability.展开更多
In this paper,we propose a regularized version of the generalized NCPfunction proposed by Hu,Huang and Chen[J.Comput.Appl.Math.,230(2009),pp.69–82].Based on this regularized function,we propose a semismooth Newton me...In this paper,we propose a regularized version of the generalized NCPfunction proposed by Hu,Huang and Chen[J.Comput.Appl.Math.,230(2009),pp.69–82].Based on this regularized function,we propose a semismooth Newton method for solving nonlinear complementarity problems,where a non-monotone line search scheme is used.In particular,we show that the proposed non-monotone method is globally and locally superlinearly convergent under suitable assumptions.We test the proposed method by solving the test problems from MCPLIB.Numerical experiments indicate that this algorithm has better numerical performance in the case of p=5 andθ∈[0.25,075]than other cases.展开更多
提出一种求解最优潮流(OPF)问题的新算法——解耦半光滑牛顿型算法.该算法是对作者的投影半光滑N ew ton算法的改进和提高,它除了保持原算法不必识别不等式约束、对界约束的特殊处理以减少讨论问题的维数等优点外,其显著的特点是结合了...提出一种求解最优潮流(OPF)问题的新算法——解耦半光滑牛顿型算法.该算法是对作者的投影半光滑N ew ton算法的改进和提高,它除了保持原算法不必识别不等式约束、对界约束的特殊处理以减少讨论问题的维数等优点外,其显著的特点是结合了电力系统固有的弱耦合性质,构造了求解OPF问题的一类解耦半光滑牛顿算法.解耦算法可达到加快计算速度、提高计算效率的目的.IEEE多个算例的数值实验以及与其他方法的比较均显示了新算法具有良好的计算效果.展开更多
对具有弱耦合特性的非线性半光滑方程组提出了牛顿型分解算法,理论上证明了新算法的收敛性.新算法享有分解法节省计算量的优点,且推广了光滑方程于半光滑方程系统.根据电力系统有功与电压、无功和相角固有的弱耦合性质,运用新算法于电...对具有弱耦合特性的非线性半光滑方程组提出了牛顿型分解算法,理论上证明了新算法的收敛性.新算法享有分解法节省计算量的优点,且推广了光滑方程于半光滑方程系统.根据电力系统有功与电压、无功和相角固有的弱耦合性质,运用新算法于电力系统的最优潮流(Optimal Power Flow-OPF)的求解,计算结果显示了算法的有效性.展开更多
基金Supported by the National Natural Science Foundation of China(No.11671183)the Fundamental Research Funds for the Central Universities(No.2018IB016,2019IA004,No.2019IB010)
文摘This paper proposes a semismooth Newton method for a class of bilinear programming problems(BLPs)based on the augmented Lagrangian,in which the BLPs are reformulated as a system of nonlinear equations with original variables and Lagrange multipliers.Without strict complementarity,the convergence of the method is studied by means of theories of semismooth analysis under the linear independence constraint qualification and strong second order sufficient condition.At last,numerical results are reported to show the performance of the proposed method.
文摘Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the system of nonsmooth equations, we develop a semismooth Newton method for the tensor complementarity problem. We prove the monotone convergence theorem for the proposed method under proper conditions.
基金Supported by the Science Technology Development Plan of Tianjin (No.06YFGZGX05600)
文摘Based on the generalized Fischer-Burmeister function, Chen et al in 2008 put forward a regularization semismooth Newton method for solving the nonlinear complementarity problem with a P0-function. In this paper, we investigate the above algorithm with the monotone line search replaced by a non-monotone line search. It is shown that the non-monotone algorithm is well-defined, and is globally and locally superlinearly convergent under standard assumptions.
文摘This paper develops and analyzes multigrid semismooth Newton methods for a class of inequality-constrained optimization problems in function space which are motivated by and include linear elastic contact problems of Signorini type. We show that after a suitable Moreau-Yosida type regularization of the problem superlinear local convergence is obtained for a class of semismooth Newton methods. In addition, estimates for the order of tile error introduced by the regularization are derived. The main part of the paper is devoted to the analysis of a multilevel preconditioner for the semismooth Newton system. We prove a rigorous bound for the contraction rate of the multigrid cycle which is robust with respect to sufficiently small regularization parameters and the number of grid levels. Moreover, it applies to adaptively refined grids. The paper concludes with numerical results.
基金supported by the Fundamental Research Fund—Shenzhen Research Institute for Big Data Startup Fund(Grant No.JCYJ-AM20190601)the Shenzhen Institute of Artificial Intelligence and Robotics for Society+2 种基金National Natural Science Foundation of China(Grant Nos.11831002 and 11871135)the Key-Area Research and Development Program of Guangdong Province(Grant No.2019B121204008)Beijing Academy of Artificial Intelligence。
文摘In this work,we present probabilistic local convergence results for a stochastic semismooth Newton method for a class of stochastic composite optimization problems involving the sum of smooth nonconvex and nonsmooth convex terms in the objective function.We assume that the gradient and Hessian information of the smooth part of the objective function can only be approximated and accessed via calling stochastic firstand second-order oracles.The approach combines stochastic semismooth Newton steps,stochastic proximal gradient steps and a globalization strategy based on growth conditions.We present tail bounds and matrix concentration inequalities for the stochastic oracles that can be utilized to control the approximation errors via appropriately adjusting or increasing the sampling rates.Under standard local assumptions,we prove that the proposed algorithm locally turns into a pure stochastic semismooth Newton method and converges r-linearly or r-superlinearly with high probability.
文摘In this paper,we propose a regularized version of the generalized NCPfunction proposed by Hu,Huang and Chen[J.Comput.Appl.Math.,230(2009),pp.69–82].Based on this regularized function,we propose a semismooth Newton method for solving nonlinear complementarity problems,where a non-monotone line search scheme is used.In particular,we show that the proposed non-monotone method is globally and locally superlinearly convergent under suitable assumptions.We test the proposed method by solving the test problems from MCPLIB.Numerical experiments indicate that this algorithm has better numerical performance in the case of p=5 andθ∈[0.25,075]than other cases.
文摘提出一种求解最优潮流(OPF)问题的新算法——解耦半光滑牛顿型算法.该算法是对作者的投影半光滑N ew ton算法的改进和提高,它除了保持原算法不必识别不等式约束、对界约束的特殊处理以减少讨论问题的维数等优点外,其显著的特点是结合了电力系统固有的弱耦合性质,构造了求解OPF问题的一类解耦半光滑牛顿算法.解耦算法可达到加快计算速度、提高计算效率的目的.IEEE多个算例的数值实验以及与其他方法的比较均显示了新算法具有良好的计算效果.
文摘对具有弱耦合特性的非线性半光滑方程组提出了牛顿型分解算法,理论上证明了新算法的收敛性.新算法享有分解法节省计算量的优点,且推广了光滑方程于半光滑方程系统.根据电力系统有功与电压、无功和相角固有的弱耦合性质,运用新算法于电力系统的最优潮流(Optimal Power Flow-OPF)的求解,计算结果显示了算法的有效性.
