Calculation of Phase Equilibria Based on the Levenberg-Marquardt Method

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.

Accelerated Matrix Recovery via Random Projection Based on Inexact Augmented Lagrange Multiplier Method

In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.

CONVERGENCE OF INEXACT CONIC NEWTON METHODS

A conic Newton method is attractive because it converges to a local minimizzer rapidly from any sufficiently good initial guess. However, it may be expensive to solve the conic Newton equation at each iterate. In this paper we consider an inexact conic Newton method, which solves the couic Newton equation oldy approximately and in sonm unspecified manner. Furthermore, we show that such method is locally convergent and characterizes the order of convergence in terms of the rate of convergence of the relative residuals.

Inexact Newton method via Lanczos decomposed technique for solving box-constrained nonlinear systems

This paper proposes an inexact Newton method via the Lanczos decomposed technique for solving the box-constrained nonlinear systems. An iterative direction is obtained by solving an affine scaling quadratic model with the Lanczos decomposed technique. By using the interior backtracking line search technique, an acceptable trial step length is found along this direction. The global convergence and the fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the results of the numerical experiments show the effectiveness of the pro- posed algorithm.

LOCAL CONVERGENCE OF INEXACT NEWTON-LIKE METHOD UNDER WEAK LIPSCHITZ CONDITIONS

The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting.We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis.The obtained results compare favorably with earlier ones such as[7,13,14,18,19].A numerical example is also provided.

Levenberg-Marquardt Method for Mathematical Programs with Linearly Complementarity Constraints

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.

An Adaptive Spectral Conjugate Gradient Method with Restart Strategy

As a generalization of the two-term conjugate gradient method(CGM),the spectral CGM is one of the effective methods for solving unconstrained optimization.In this paper,we enhance the JJSL conjugate parameter,initially proposed by Jiang et al.(Computational and Applied Mathematics,2021,40:174),through the utilization of a convex combination technique.And this improvement allows for an adaptive search direction by integrating a newly constructed spectral gradient-type restart strategy.Then,we develop a new spectral CGM by employing an inexact line search to determine the step size.With the application of the weak Wolfe line search,we establish the sufficient descent property of the proposed search direction.Moreover,under general assumptions,including the employment of the strong Wolfe line search for step size calculation,we demonstrate the global convergence of our new algorithm.Finally,the given unconstrained optimization test results show that the new algorithm is effective.

Improved hybrid iterative optimization method for seismic full waveform inversion

In full waveform inversion （FWI）, Hessian information of the misfit function is of vital importance for accelerating the convergence of the inversion; however, it usually is not feasible to directly calculate the Hessian matrix and its inverse. Although the limited memory Broyden-Fletcher-Goldfarb-Shanno （L-BFGS） or Hessian-free inexact Newton （HFN） methods are able to use approximate Hessian information, the information they collect is limited. The two methods can be interlaced because they are able to provide Hessian information for each other; however, the performance of the hybrid iterative method is dependent on the effective switch between the two methods. We have designed a new scheme to realize the dynamic switch between the two methods based on the decrease ratio （DR） of the misfit function （objective function）, and we propose a modified hybrid iterative optimization method. In the new scheme, we compare the DR of the two methods for a given computational cost, and choose the method with a faster DR. Using these steps, the modified method always implements the most efficient method. The results of Marmousi and overthrust model testings indicate that the convergence with our modified method is significantly faster than that in the L-BFGS method with no loss of inversion quality. Moreover, our modified outperforms the enriched method by a little speedup of the convergence. It also exhibits better efficiency than the HFN method.

GLOBAL COVERGENCE OF THE NON-QUASI-NEWTON METHOD FOR UNCONSTRAINED OPTIMIZATION PROBLEMS

In this paper, the non-quasi-Newton＇s family with inexact line search applied to unconstrained optimization problems is studied. A new update formula for non-quasi-Newton＇s family is proposed. It is proved that the constituted algorithm with either Wolfe-type or Armijotype line search converges globally and Q-superlinearly if the function to be minimized has Lipschitz continuous gradient.

GLOBAL CONVERGENCE OF A TRUST REGION ALGORITHM USING INEXACT GRADIENT FOR EQUALITY-CONSTRAINED OPTIMIZATION

A trust-region algorithm is presented for a nonlinear optimization problem of equality-constraints. The characterization of the algorithm is using inexact gradient information. Global convergence results are demonstrated where the gradient values are obeyed a simple relative error condition.

一种求解约束非线性方程组的非精确Levenberg-Marquardt算法

利用绝对值函数的光滑函数将约束非线性方程组转化为一个光滑方程组,用非精确Levenberg-Mar-quardt方法求解该光滑方程组,得到一种求解约束非线性方程组的非精确Levenberg-Marquardt算法,证明该算法具有全局收敛性,并给出数值实验.

GLOBAL CONVERGENCE OF THE GENERAL THREE TERM CONJUGATE GRADIENT METHODS WITH THE RELAXED STRONG WOLFE LINE SEARCH

The global convergence of the general three term conjugate gradient methods with the relaxed strong Wolfe line search is proved.

An Inexact Restoration Package for Bilevel Programming Problems

Bilevel programming problems are a class of optimization problems with hierarchical structure where one of the con-straints is also an optimization problem. Inexact restoration methods were introduced for solving nonlinear programming problems a few years ago. They generate a sequence of, generally, infeasible iterates with intermediate iterations that consist of inexactly restored points. In this paper we present a software environment for solving bilevel program-ming problems using an inexact restoration technique without replacing the lower level problem by its KKT optimality conditions. With this strategy we maintain the minimization structure of the lower level problem and avoid spurious solutions. The environment is a user-friendly set of Fortran 90 modules which is easily and highly configurable. It is prepared to use two well-tested minimization solvers and different formulations in one of the minimization subproblems. We validate our implementation using a set of test problems from the literature, comparing different formulations and the use of the minimization solvers.

Calibration Method of Magnetometer Based on BP Neural Network

Due to the influence of processing technology and environmental factors, there are errors in attitude measurement with the three-axis magnetometer, and the change of parameters during the operation of the magnetometer in orbit will have a great impact on the measurement accuracy. This paper studies the calibration method of magnetometer based on BP neural network, which reduces the influence of model error on calibration accuracy. Firstly, the error model of the magnetometer and the structural characteristics of the BP neural network are analyzed. Secondly, the number of hidden layers and hidden nodes is optimized. To avoid the problem of slow convergence and low accuracy of basic BP algorithm, this paper uses the Levenberg Marquardt backpropagation training method to improve the training speed and prediction accuracy and realizes the on-orbit calibration of magnetometer through online training of the neural network. Finally, the effectiveness of the method is verified by numerical simulation. The results show that the neural network designed in this paper can effectively reduce the measurement error of magnetometer, while the online training can effectively reduce the error caused by the change of magnetometer parameters, and reduce the measurement error of magnetometer to less than 10 nT.

An Inexact Affine Scaling Levenberg-Marquardt Method Under Local Error Bound Conditions

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.

水平线性互补问题的一种非精确光滑牛顿算法

为了提高求解水平线性互补问题的效率,本文利用一种光滑函数,将水平线性互补问题转化为与之等价的光滑方程组,采用非精确牛顿法求解该方程组,得到了水平线性互补问题的一种非精确光滑牛顿算法.在适当的条件下证明了该算法的适定性和局部二阶收敛性,数值实验表明该算法稳定有效.

一种非精确非光滑信赖域算法

Aravkin等人提出了求解非光滑优化问题min_(x∈R^(d))f(x)+h(x)的非光滑信赖域算法(采用f的精确梯度),其中f是连续可微函数,h是邻近有界且下半连续的真函数。文章研究当该问题中f:=1/n ∑_(i=1)^(n)f_(i)(n很大且每个分量函数fi是连续可微)时,求解这类大规模可分离非光滑优化问题的有效算法。结合非精确算法和非光滑信赖域算法的思想,提出了用非精确梯度代替精确梯度的非精确非光滑信赖域算法。与非光滑信赖域算法(采用精确梯度)相比,该算法降低了每次迭代的计算量。在一定的假设条件下,证明了算法的迭代复杂度。

对等控制孤岛微电网的静态安全风险评估

间歇性可再生能源发电出力的随机性、波动性及缺乏主网的支撑,增大了孤岛微电网系统的安全稳定运行风险,由此有必要对其进行静态安全性评估。安全分析准则和潮流计算是电力系统静态安全风险评估的基础。针对对等控制孤岛微电网的特性,提出结合N-1事故与解列方案的静态安全分析准则;并计及故障解列后孤岛系统的无平衡节点特性及存在的不确定性因素,提出基于LMIL(Levenberg Marquardt method with inexact line-search)算法的随机潮流模拟法,计算解列孤岛系统的随机潮流,进而得到考虑频率越限风险的静态安全风险指标的概率分布,以更全面地评估系统的静态安全水平。以17节点孤岛微电网测试系统为算例,验证了所提评估方法的正确性和有效性。

基于预条件处理GMRES的不精确牛顿法潮流计算

结合大规模电力系统修正方程组高维超稀疏性以及短向量的特点,提出以Krylov子空间方法研究电力系统方程计算问题。针对牛顿法潮流计算,采用预条件处理的GMRES方法求解高维稀疏的修正方程组,提出一种完整的基于预条件处理GMRES的不精确牛顿潮流算法,设计实现不同的预条件子,并以此为基础详细比较各类预条件子的预处理效果。通过对IEEE30、IEEE118和多个合成的大规模电力系统进行潮流计算,结果表明ILU预条件子比其他预条件子需要更少的迭代次数和浮点运算次数,当系统规模达到3000节点左右时,基于ILU预条件子的不精确牛顿法与传统的LU直接分解法相比,浮点运算次数减少了50%,内存使用量减少了将近10%,并且随着系统规模的增大,浮点运算次数基本上保持在LU直接法的50%左右,对大规模电力系统的潮流计算极为有利。

SOLVERS FOR SYSTEMS OF LARGE SPARSE LINEAR AND NONLINEAR EQUATIONS BASED ON MULTI-GPUS

Numerical treatment of engineering application problems often eventually results in a solution of systems of linear or nonlinear equations.The solution process using digital computational devices usually takes tremendous time due to the extremely large size encountered in most real-world engineering applications.So,practical solvers for systems of linear and nonlinear equations based on multi graphic process units（GPUs）are proposed in order to accelerate the solving process.In the linear and nonlinear solvers,the preconditioned bi-conjugate gradient stable（PBi-CGstab）method and the Inexact Newton method are used to achieve the fast and stable convergence behavior.Multi-GPUs are utilized to obtain more data storage that large size problems need.