拉格朗日问题

The performance function and constraint equation in calculus of variations were constructed by analyzing input and output equations of the system containing parameters to be estimated. The traditional parameter estimation problem could be transformed to that of least squares estimation.

通过分析包含待估参数的系统输入输出方程，构造变分法中的性能范函和约束方程，将传统的参数估计问题化为带约束条件的最小二乘估计问题；利用拉格朗日乘子法，分析参数估计方程，得到参数的最优估计。

State variables at left end of time interval and costate at right end of time interval were taken as independent mixed variables, and the state and costate variables inside the time interval were approximated by Lagrange interpolation. Nonlinear optimal control problems were replaced by nonlinear equations through dual variable principle, and the symplectic algorithm for solving nonlinear optimal control problems can be obtained at the same time.

以时间区段一端状态和另一端协态作为混合独立变量，在时间区段内采用拉格朗日插值近似状态变量与协态变量，然后利用对偶变量变分原理并将非线性最优控制问题转化为非线性方程组的求解，最终得到求解非线性动力学系统最优控制问题的保辛数值方法。

State variables at left end of time interval and costate at right end of time interval were taken as independent mixed variables,and the state and costate variables inside the time interval were approximated by Lagrange interpolation.

以时间区段一端状态和另一端协态作为混合独立变量，在时间区段内采用拉格朗日插值近似状态变量与协态变量，然后利用对偶变量变分原理并将非线性最优控制问题转化为非线性方程组的求解，最终得到求解非线性动力学系统最优控制问题的保辛数值方法。

A deterministic global optimization algorithm based on Lagrangian dual is proposed for solving the nonconvex quadratic programming with nonconvex quadratic constraints, which is often encountered in technical design and operational research.

针对带有非凸二次函数约束的非凸二次规划问题，提出了一个基于拉格朗日对偶的确定型全局优化算法，这类优化算法可广泛应用于工程设计和非线性系统的鲁棒稳定性分析等实际问题中。

The sub-gradient algorithm is designed to solve lagrangian dual problem according to its characteristic.

根据问题的特性设计了次梯度算法求解拉格朗日对偶问题。

On the basis of analyzing detailedly the correlative conclusions of the extreme value problem of two variable Function , this paper extends the essential condition and sufficient condition to multi-variable function by using the Taylor Formula 、Quadratic Form 、Hesse Matrix and so on,and gives a method to judge the extreme value of point whose vice-derivative doesn't exist by utilizing Lagranges theorem ,so that compensates for the blank above ,which turns idea that solve the extreme value of multi-variable function problem one by one into realities.

摘要本文在对二元函数极值问题的相关结论进行分析的基础上，用泰勒公式、二次型、海赛矩阵作为工具，将二元函数极值的必要条件和充分条件推广到多元函数的情形，而且还利用拉格朗日中值定理找到了一个判别不可导点处极值的方法，从而弥补了多元函数不可导点处极值无法判断的空缺。因而，使得用穷举判断的方法来解决多元函数的极值问题这一思路成为现实。

So in this paper, first, the ELM is improved by taking the interpolation process under absolute coordinates, so that the horizontal interpolation could be taken in a horizontal plain of the same altitude, thus the problem of unreasonable interpolation sources is prevented.

所以本文首先对欧拉-拉格朗日方法作了改进，把其中的插值过程改在绝对坐标下进行，使得水平插值过程在同一高度的水平面内进行，从而克服了插值不合理的问题。

Based on the elliptic curve cryptosystem, a multi-key sharing scheme in which the Lagrange interpolation polynomial is used and the cheaters can be detected is presented.

文章利用拉格朗日插值的思想，提出了一种基于椭圆曲线的可防欺诈的动态多密钥共享方案；方案中每个参与者的子密钥可以不受限制的重复使用，因此在更新主密钥的时候，无需更改参与者的子密钥，从而减少了可信中心与参与者之间的通信量；方案可灵活地增删参与者，且安全性是基于椭圆曲线离散对数问题的难解性，因而其安全性比在有限域上更高；方案实现过程中解决了检验子密钥真伪的问题。

Based on investigating the short and intermittent structural plane, surrounding rockmass of the huge underground powerhouse in Three Gorge has been divided into five statistical similar areas according to its characteristic of rockmass structure by applying structure plane network simulation. Distributed characteristic of structural plane has been studied and based on it mechanics macrocharacteristic has been studied:(1) It is the first time that based on the result of structural plane network simulation rockmass quality of every structure area has been studied with applying blurred information optimum seeking technique and be compared with and RQD method. The result indicated that it has better effect to evaluate rock quality with blurred information optimum seeking technique.(2) It is the first time that the interrelated relationship of the result of in-suit rockmass deformation test and fractal characteristic of structure on test point has been applied to determine rockmass deformation parameter. The question of evaluating deformation parameters of engineering rockmass has been solved.(3) Anisotropy strength parameters have been determined by applying shear zones simulating method. The result indicated that there are different strength parameters in every structural area, after comparing the result of Fast Lagranian Analysis in continua number simulation method with that of method of weighted mean according to continuity. The result indicated that when a suitable reduction coefficient is been considered rockmass strength parameters can be simply and fast determined with method of weighted mean according to continuity where there is not enough test result.(4) Damage characteristic has been studied based on analysis fractal characteristic of structural plane network by applying fractal-damage coupling method. It is the first time that the concept of damage degree has been set up to evaluate rockmass quality according to the biggest principal damage.

通过对短小、断续性结构面统计特征的研究，将地下厂房区划分为五个岩体结构统计均质区，应用结构面网络模拟技术研究了各个岩体结构均质区结构面的发育分布特征，并以此为基础对岩体宏观力学特性进行了深入研究：(1)首次通过在结构面网络模拟计算结果的基础上，应用模糊信息优化技术研究了厂房区各个岩体结构分区的岩体质量并与岩体基本质量指标研究法、岩体质量指标RQD值研究法作了对比，结果表明以结构面网络模拟结果为基础，应用模糊信息优化技术进行岩体质量评价能够取得很好的效果，解决了工程岩体质量的定量评价问题；(2)首次通过对已有的现场大型岩体变形试验结果的分析及相应试验点岩体结构分形特征的研究，建立了结构面分布的分形维数和岩体变形指标间的相关关系，并首次应用这种相关关系根据结构面网络模拟及分形维数计算结果确定了缺少试验结果的地下厂房各工程部位的岩体变形指标而且应用于围岩稳定性计算，解决了工程岩体宏观变形参数的评价问题；(3)应用在结构面模拟网络图上模拟岩体剪切破坏带的方法，得出了岩体在不同方向上的强度参数，结果表明岩体的强度特性与结构面的发育方向和程度有着明显的关系，各个岩体结构均质区具有不同的强度参数；同时还通过应用拉格朗日差分法对岩体强度指标的数值分析并与按连通率加权平均计算法对比表明，按连通率加权平均计算法在考虑合理的折减系数情况下，可以简单快速地对缺少现场大型试验结果的具体工程部位实现岩体强度指标的估计，通过以上研究解决了工程岩体抗剪强度参数的评价问题；(4)通过结构面网络分形维数的研究，应用分形损伤耦合分析方法，研究了厂房区岩体的损伤特征，并首次引入"损伤度"的概念，建立了应用最大主损伤评价岩体质量的指标体系。

Mathematical analysis and experimental results show that this adaptive total variation denoising algorithm can solve the problem that Euler-Lagrange equation of the traditional total variation is not well posed, improve the deficiencies of the slow denosing speed, and reduce the step effect and noise leaving out resulting from using the traditional total variation.

数学分析和实验证明，论文提出的变分去噪模型克服了传统的总体变分模型的欧拉-拉格朗日方程非适定问题，改善了传统总体变分模型去噪速度慢、有阶梯效应、噪声遗漏等缺陷。

For a big chunk of credit-card losses; the number of filings (and thus charge-off rates) would be rising again, whether

年美国个人破产法的一个改动使得破产登记急速下降，而后引起了信用卡大规模的亏损。

Eph. 4:23 And that you be renewed in the spirit of your mind

弗四23 而在你们心思的灵里得以更新