The Levenberg-Marquardt method, the best algorithm to obtain the least-square solution of nonlinear equations, is applied to calculate the stable phase equilibria. It can get the best combination between robustness an...The Levenberg-Marquardt method, the best algorithm to obtain the least-square solution of nonlinear equations, is applied to calculate the stable phase equilibria. It can get the best combination between robustness and speed of the calculations. Its application to ternary AI-Si-Mg system is executed in detail. The calculated phase equilibria agree well with the experimental results. Furthermore, the Levenberg-Marquardt method is not sensitive to the initial values.展开更多
A multi-constituent water quality model is presented,Which relates carbonaceous biochemical oxygen demand (CBOD),amonia (NH3-N), nitrite(NO2-N), nitrate(NO3-N) and dissolvedoxygen(DO). The parameters are solved by Mar...A multi-constituent water quality model is presented,Which relates carbonaceous biochemical oxygen demand (CBOD),amonia (NH3-N), nitrite(NO2-N), nitrate(NO3-N) and dissolvedoxygen(DO). The parameters are solved by Marquardt Method (i. e.,Dampled Least Square Method) while initial values inoptimization are produced by Monte-Carlo Method. The Potential ofthe method as a parameter estimation aid is demonstrated for theapplication to the Liangyi Rver, JiangSu Province of China and by aspecial comparison with Gauss Method.展开更多
In this paper, a new method for solving a mathematical programming problem with linearly complementarity constraints (MPLCC) is introduced, which applies the Levenberg-Marquardt (L-M) method to solve the B-stationary ...In this paper, a new method for solving a mathematical programming problem with linearly complementarity constraints (MPLCC) is introduced, which applies the Levenberg-Marquardt (L-M) method to solve the B-stationary condition of original problem. Under the MPEC-LICQ, the proposed method is proved convergent to B-stationary point of MPLCC.展开更多
A parameter-self-adjusting Levenberg-Marquardt method （PSA-LMM） is proposed for solving a nonlinear system of equations F（x） = 0, where F ：R^n→R^n is a semismooth mapping. At each iteration, the LM parameter μk...A parameter-self-adjusting Levenberg-Marquardt method （PSA-LMM） is proposed for solving a nonlinear system of equations F（x） = 0, where F ：R^n→R^n is a semismooth mapping. At each iteration, the LM parameter μk is automatically adjusted based on the ratio between actual reduction and predicted reduction. The global convergence of PSA- LMM for solving semismooth equations is demonstrated. Under the BD-regular condition, we prove that PSA-LMM is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations. Numerical results for solving nonlinear complementarity problems are presented.展开更多
In this paper,we investigate the global complexity bound for the inexact Levenberg–Marquardt method,where the Jacobian may be perturbed and the solution is possibly not exact.Under reasonable assumptions,we show that...In this paper,we investigate the global complexity bound for the inexact Levenberg–Marquardt method,where the Jacobian may be perturbed and the solution is possibly not exact.Under reasonable assumptions,we show that the global complexity bound is O(ε^(−2)),which is the same as the exact case.We also show that it can be reduced to O(lgε^(−1))under some regularity assumption.展开更多
In this paper, we propose an extended Levenberg-Marquardt （ELM） framework that generalizes the classic Levenberg-Marquardt （LM） method to solve the unconstrained minimization problem min ρ（r（x））, where r ： R...In this paper, we propose an extended Levenberg-Marquardt （ELM） framework that generalizes the classic Levenberg-Marquardt （LM） method to solve the unconstrained minimization problem min ρ（r（x））, where r ： Rn→ Rm and ρ ： Rm → R. We also develop a few inexact variants which generalize ELM to the cases where the inner subproblem is not solved exactly and the Jaeobian is simplified, or perturbed. Global convergence and local superlinear convergence are established under certain suitable conditions. Numerical results show that our methods are promising.展开更多
We propose an inexact affine scaling Levenberg-Marquardt method for solving bound-constrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition...We propose an inexact affine scaling Levenberg-Marquardt method for solving bound-constrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition. The affine scaling Levenberg-Marquardt equation is based on a minimization of the squared Euclidean norm of linearized model adding a quadratic affine scaling matrix to find a solution which belongs to the bounded constraints on variable. The global convergence and the superlinear convergence rate are proved.Numerical results show that the new algorithm is efficient.展开更多
基金This research is supported by the State Key Fundamental Research Project(G2000067202-1).
文摘The Levenberg-Marquardt method, the best algorithm to obtain the least-square solution of nonlinear equations, is applied to calculate the stable phase equilibria. It can get the best combination between robustness and speed of the calculations. Its application to ternary AI-Si-Mg system is executed in detail. The calculated phase equilibria agree well with the experimental results. Furthermore, the Levenberg-Marquardt method is not sensitive to the initial values.
文摘A multi-constituent water quality model is presented,Which relates carbonaceous biochemical oxygen demand (CBOD),amonia (NH3-N), nitrite(NO2-N), nitrate(NO3-N) and dissolvedoxygen(DO). The parameters are solved by Marquardt Method (i. e.,Dampled Least Square Method) while initial values inoptimization are produced by Monte-Carlo Method. The Potential ofthe method as a parameter estimation aid is demonstrated for theapplication to the Liangyi Rver, JiangSu Province of China and by aspecial comparison with Gauss Method.
文摘In this paper, a new method for solving a mathematical programming problem with linearly complementarity constraints (MPLCC) is introduced, which applies the Levenberg-Marquardt (L-M) method to solve the B-stationary condition of original problem. Under the MPEC-LICQ, the proposed method is proved convergent to B-stationary point of MPLCC.
基金Acknowledgments. This work is supported by the National Natural Science Foundation of China under projects Nos. 11071029, 11101064 and 91130007 and speciMized Research Fund for the Doctoral Program of Higher Education （20110041120039）. We are grateful to the associate editor and anonymous referee＇s comments to improve the quality of the manuscript. The second author also appreciate the discussion with his student Miao Xiaonan.
文摘A parameter-self-adjusting Levenberg-Marquardt method （PSA-LMM） is proposed for solving a nonlinear system of equations F（x） = 0, where F ：R^n→R^n is a semismooth mapping. At each iteration, the LM parameter μk is automatically adjusted based on the ratio between actual reduction and predicted reduction. The global convergence of PSA- LMM for solving semismooth equations is demonstrated. Under the BD-regular condition, we prove that PSA-LMM is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations. Numerical results for solving nonlinear complementarity problems are presented.
基金This work was partially supported by the National Natural Science Foundation of China(No.11571234).
文摘In this paper,we investigate the global complexity bound for the inexact Levenberg–Marquardt method,where the Jacobian may be perturbed and the solution is possibly not exact.Under reasonable assumptions,we show that the global complexity bound is O(ε^(−2)),which is the same as the exact case.We also show that it can be reduced to O(lgε^(−1))under some regularity assumption.
文摘In this paper, we propose an extended Levenberg-Marquardt （ELM） framework that generalizes the classic Levenberg-Marquardt （LM） method to solve the unconstrained minimization problem min ρ（r（x））, where r ： Rn→ Rm and ρ ： Rm → R. We also develop a few inexact variants which generalize ELM to the cases where the inner subproblem is not solved exactly and the Jaeobian is simplified, or perturbed. Global convergence and local superlinear convergence are established under certain suitable conditions. Numerical results show that our methods are promising.
基金Supported by National Natural Science Foundation of China(No.11571074)Scientific Research Fund of Hunan Provincial Education Department(No.18A351,17C0393)Natural Science Foundation of Hunan Province(No.2019JJ50105)
文摘We propose an inexact affine scaling Levenberg-Marquardt method for solving bound-constrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition. The affine scaling Levenberg-Marquardt equation is based on a minimization of the squared Euclidean norm of linearized model adding a quadratic affine scaling matrix to find a solution which belongs to the bounded constraints on variable. The global convergence and the superlinear convergence rate are proved.Numerical results show that the new algorithm is efficient.
