拉格朗日乘数法在经济学中的应用

介绍了拉格朗日乘数法在经济学中的应用,举例说明利用拉格朗日乘数法解决效用最大化问题和成本最小化问题,并且说明在经济学中拉格朗日乘数λ也具有一定的经济含义。     

拉格朗日乘数求大转盘处红绿灯最优周期

本文用拉格朗日乘数法解决大转盘处红绿灯的最优周期问题。     

最优温度调节在温度控制系统中的应用

利用变分方法,采用拉格朗日乘数法,通过构造哈密顿函数来确定最优温度轨线的方法,研究了温度控制系统中的温度调节问题,给出了达到理想温度时的流体流量的调节律。     

三元函数最值问题的解法探究

多元函数最值求解问题是近几年高考考查热点难点,利用常用不等式法、柯西不等式法、拉格朗日乘数法、空间解析几何法、向量法、三角换元法等多种方法、多个角度分析解决多元函数最值问题,帮助学生更全面地掌握此类问题。     

深圳市固定资产投资与经济增长的辩证分析

选用深圳统计年鉴1979年~2012年(34个)固定资产投资与生产总值的年度数据,应用EViews 7.0,对深圳市固定资产投资与经济增长进行辩证分析。发现固定资产投资与GDP存在显著的长期协整关系,GDP增长是固定资产投资增加的原因,估计模型并用拉格朗日乘数法对模型进行修正,同时提出相应的建议。     

马可维兹模型有效边界分

本文通过计算10支股票的期望收益率、标准差、协方差矩阵和相关系数,用拉格朗日乘数法对马可维兹模型(MM模型)进行最优投资组合的选择与分析,分可以卖空和不可以卖空绘制MM模型的有效边界图,更好地确定投资对象及最优分配比例,以便实现收益最大、风险最小的目标。     

问题与解答

拉格朗日点是如何被测到的?——河南省郑州市河南财经学院西校区张经纬拉格朗日点于1772年由法国数学家拉格朗日推算得出,指的是在两大物体引力作用下,能使小物体稳定的点。1906年,天文学家发现了第588号小行星在木星和太阳的作用下,正好处在拉格朗日点上,它同木星几乎在同一轨道上超前60°运动,这三个星体一起构成运动着的等边三角形。同年发现的第617号小行星也在木星轨道上落后60°左右,构成第2个拉格朗日正三角形。这就是早期被观测到的拉格朗日点。人们还发现在宇宙中拉格朗日点普遍存在。具体参看我刊2003年第02期《拉格朗日点的妙用》一文。     

角点柱支承矩形板的设计计算

通过选取适当的位称函数,同时利用拉格朗日乘子法给出了角点柱式支承矩形板弯曲问题有的内力及位移的计算公式,该法计算精度高,与前人方法相比,计算了过程简单。     

一类多目标规划问题的鞍点判别法

本文主要研究了向量最小值这一类多目标规划问题,并通过引入α-invex函数,α-η-拉格朗日函数的概念,给出了这类多目标规划问题的鞍点判别法。     

拉格朗日乘数法求距离

利用条件极值的拉格朗日乘数法求解点到直线、曲线和曲面的最小距离。     

Parallel computation approaches for flexible multibody dynamics simulations

Finite element based formulations for flexible multibody systems are becoming increasingly popular and as the complexity of the configurations to be treated increases, so does the computational cost. It seems natural to investigate the applicability of parallel processing to this type of problems; domain decomposition techniques have been used extensively for this purpose. In this approach, the computational domain is divided into non-overlapping sub-domains, and the continuity of the displacement field across sub-domain boundaries is enforced via the Lagrange multiplier technique. In the finite element literature, this approach is presented as a mathematical algorithm that enables parallel processing. In this paper, the divided system is viewed as a flexible multibody system, and the sub-domains are connected by kinematic constraints. Consequently, all the techniques applicable to the enforcement of constraints in multibody systems become applicable to the present problem. In particular, it is shown that a combination of the localized Lagrange multiplier technique with the augmented Lagrange formulation leads to interesting solution strategies.     

Consensus-based distributed fixed-time optimization for a class of resource allocation problems

A class of resource allocation problems with equality constraint are considered in this paper, such as economic dispatch problem in smart grid systems, which is essentially an optimization problem. Inspired by the Lagrange multiplier method, the resource allocation problem is transformed into a multi-agent consensus problem for large-scale networked distributed nodes. A consensus-based distributed fixed-time optimization algorithm is presented, where the information exchange network is depicted by a strongly connected and weight-balanced digraph. This type of communication network can ensure that the equality constraint always holds. Moreover, a new globally fixed-time stability theorem for nonlinear systems is first given in this paper. Based on this theorem and consensus theory, the optimal resource allocation scheme can be given in a fixed time. Finally, the application and comparison of the designed algorithm show that the algorithm can effectively solve the allocation problem of power resources such as economic dispatch.     

Free final time fractional optimal control problems

A formulation and solution scheme of free final time fractional optimal control problems is presented in this paper. The dynamic constraint is described by a fractional differential equation. Performance index considered is a function of both the state and control variables. The necessary conditions of optimality and the transversality condition are obtained using Lagrange multiplier technique. A numerical technique similar to Shooting method is used for solving the optimal conditions. Numerical example is provided to show the effectiveness of the formulation and numerical solution scheme. It is interesting to note that the final time changes with the interchange of the boundary conditions, which does not occur in classical optimal control problems.     

对投资乘数的应用分析

投资乘数原理是凯恩斯学派运用数学模型从数量上描述投资需求对经济增长影响关系的方法，具有一定的借鉴作用。但在实际投资活动中，一笔投资并非按设定的乘数引起收入的倍增，甚至会有很大的差异。本文仅就投资乘数的具体应用作出分析，以揭示其理论与实际应用之差异。     

The complex adjoint approaches to network sensitivities

We present a comprehensive comparison between the widely used Lagrange multiplier and Tellegen's theorem approaches to sensitivity calculations in electrical networks. The two approaches are described on a unified basis using the conjugate notation. Different aspects of comparison can thereby be investigated. The linear electronic circuit analysis case is seen to be a special case. Analytical and numerical results are shown for 2-bus and 6-bus power systems.     

Cooperative output feedback control of a mobile dual flexible manipulator

This paper addresses the cooperative output feedback control of a mobile dual flexible manipulator, which is mounted at a moving platform to grasp and move a rigid object. We derive the distributed parameter model with geometric constraints for the dual flexible manipulator system by utilizing the Lagrange multiplier method and the Hamilton’s principle, which avoids the problem of control spillover. This paper considers a case where the states of system are difficult to measure directly and exploits the high gain observer theory to design the state observers for estimating the unavailable states. Then the cooperative output feedback control scheme is developed by the Lyapunov’s method, which enables the cooperative control of the flexible manipulator system. Furthermore, under the cooperative output feedback control scheme, we prove that the states of the system are uniformly bounded. Finally, the feasibility of the designed cooperative output feedback controllers is verified by numerical simulation.     

Application of a parametrization method to problem of optimal control

A new approach to the problem of optimal control of linear dynamic systems is proposed that makes use of a method of input and state parametrization to transform the original problem into a problem of the Calculus of Variations. In contrast to the standard approaches for this class of problems, two salient features of the new approach are that no Lagrange multiplier functions need to be invoked and that the class of inputs can be restricted to the - relatively small - class of continuous functions, even for problems with fixed end-states. The resulting necessary conditions of optimality, i.e., the Euler-Lagrange equation and the boundary conditions for the transformed problem, are proved to be equivalent to those resulting from the standard method of First Variations. In case of quadratic cost functionals, the new approach provides a simpler alternative to the well known, but equally difficult, Riccati differential equation approach and results in a simple dynamic state-feedback implementation of the optimal control.     

浅谈拉格朗日中值定理的推广和应用

拉格朗日中值定理是微分学中的重要的基本定理之一,也是三大微分中值定理中的核心定理,本文应用拉格朗日中值定理及推论证明等式、举例说明Lagrange中值定理在求解极限中的应用、就拉格朗日中值定理的一个推广进行了浅要说明,其中在拉格朗日中值定理推广上证明了拉格朗日中值定理在开区间有连续右导数的情况也能使用,这一推广大大拓宽了拉格朗日中值定理的使用范围。     

考虑权重乘子的核心制造商风险动态评估研究——基于供应链视角

文章结合FMECA分析的风险诊断方法、C-POWA算子中在区间数信息集结的算法步骤以及权重乘子在动态评价的应用,提出了制造业供应链核心制造商风险动态评估模型。文章以单一供应商、单一制造商与单一分销商组成的三级供应链运作模型为基础,应用流程图法分析制造业供应链中核心制造商运作环节所面临的风险因素,应用FMECA分析法诊断风险因素属性指标值大小,依据C-POWA算子在多序列属性区间数信息集结的算法步骤对属性区间指标值进行集结,应用权重乘子确定评估指标动态权重。最后,以实际算例对上述过程进行验算分析,以此说明该评估模型的可行性与有效性。