拉格朗日问题

From the perspective of the variational discussion of Lagrange\'s contributions to the principles and analyzation of mechanics,the interaction between the calculus of variations and mechanics,especially the variational principles is made clear.

结合拉格朗日微积分代数化方案和18世纪可积性条件的研究，探讨了拉格朗日关于变分法基础研究的数学背景及相关工作；通过对《分析力学》中静力学约束平衡问题的细致考察，系统探讨了变分法中乘子法则的力学渊源、提出过程及意义。

Firstly, the approach formulates a cost functional to turn the inverse problem into a constrained minimization problem according to least squares criterion, then the resulting constrained minimization problem is transformed into an unconstrained minimization problem by using a penalty function technique, and then the closed Fr chet derivatives of the Lagrange function with respect to the properties are derived based on the calculus of variations, finally, one can solve the resulting problem by using any gradient-based algorithm and the finite-difference time-domain method.

该方法首先以最小二乘准则构造目标函数，将逆问题表示为约束最小化问题；接着应用罚函数法转化为无约束最小化问题；然后基于变分计算导出闭式的拉格朗日函数关于特征参数的Fr chet导数；最后借助梯度算法和时域有限差分法迭代反演德拜模型参数。

In this paper,the author introduced an approximate augmented Lagrangian function in nonlinear programming,established dual mapping and the related dual problem of this augmented Lagrangian function and obtained the results of approximate strong duality and approximate weak duality of original problems.

介绍了非线性规划中的一种近似增广拉格朗日函数，建立了基于这种增广拉格朗日函数的对偶映射和相应的对偶问题，得到了原问题和对偶问题的强近似对偶和弱近似对偶结果。

For the single objective optimization, the decomposition and coordination method is adopted to build the decomposition and coordination model according to the existing sub-area division conditions of power networks. Then using the Augmented Lagrange method, the minimization problem of decomposition and coordination model can be changed to the saddle point problem of augmented Lagrangian function. Finally, the so called auxiliary problem principle is selected to decompose variables as well as the functions. This transforms the voltage and reactive optimization problem of the wholenetworks to some sub-problems in some sub-areas.

对于单目标无功电压优化，根据实际电网分区情况，采用分解协调法复制各分区的边界节点，建立分解协调模型，采用增广拉格朗日法将求分解协调模型的极小值问题转化为求增广拉格朗日函数的鞍点问题，然后采用辅助问题原理分解变量和增广拉格朗日函数，从而将全网无功电压优化问题分解为多个分区的分布式并行优化问题。

We establish a Lagrange multiplier theorem for strict efficiency in convex settings and express strict points as saddle points of an appropriate Lagrangian function.

讨论凸多目标最优化问题的严有效解，建立了拉格朗日乘子定理，并把严有效解表示为一个适当的拉格朗日函数的鞍

In this paper,some approximate optimal solutions and an augmented Lagrangian function in nonlinear programming were introduced,duality map and duality problems based on the augmented Lagrangian function were established, relationship between the approximate optimal solutions of augmented Lagrangian function and primal problem was discussed.

介绍了几种近似最优解和增广拉格朗日函数，建立了基于增广拉格朗日函数的对偶映射和相应的对偶问题，讨论了增广拉格朗日函数的几种近似解和原问题的几种近似解的关系，得到的结果推广了一些已有的结论。

Therefore, from the theoretical point of view, a solution of the constrained problem and the corresponding values of the Lagrange multipliers can be found not only by the well known method of multipliers but also by performing a single unconstrained minimization of the Hestenes-Powell augmented Lagrangian function on the product space of problem variables and multipliers.

因此，从理论的观点来看，原约束问题的解和对应的拉格朗日乘子值不仅可以用众所周知的乘子法求得，而且可以通过对Hestenes-Powell增广拉格朗日函数在原问题变量和乘子变量的积空间上执行一个单一的无约束极小化来获得。

Under suitable assumptions, the relationship is established between the unconstrained minimization of the Hestenes-Powell augmented Lagrangian function on the space of problem variables and the solution of the original constrained problem, and a relationship is also presented between the unconstrained minimization of the Hestenes-Powell augmented Lagrangian function on the product space of problem variables and multipliers and the solution of the original constrained problem.

在适当的条件下，我们建立了Hestenes-Powell增广拉格朗日函数在原问题变量空间上的无约束极小与原约束问题的解之间的关系，并且也给出了Hestenes-Powell增广拉格朗日函数在原问题变量和乘子变量的积空间上的无约束极小与原约束问题的解之间的一个关系。

Probability distribution of anchoring data can be regarded as nonlinear programming with certain restriction. Programming with the restriction can be transferred to nonlinear programming without restriction by use of Lagrange Method and can be then easily resolved by step-acceleration method of one variable.

锚固强度参数的概率分布问题可以归结为带约束的非线性规划问题，利用条件极值的拉格朗日乘数法可以将带约束的规划问题转化为无约束的非线性规划问题，并可以采用运筹学中的单变量步长加速法非常方便地求解。

In this paper, the Hestenes-Powell augmented Lagrangian function is again considered, for solving equality constrained problems via unconstrained minimization techniques.

在适当的条件下，我们建立了Hestenes-Powell 增广拉格朗日函数在原问题变量空间上的无约束极小与原约束问题的解之间的关系，并且也给出了Hestenes-Powell 增广拉格朗日函数在原问题变量和乘子变量的积空间上的无约束极小与原约束问题的解之间的一个关系。

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