对等式约束非线性规划问题的Hestenes-Powell增广拉格朗日函数的进一步研究

本文对用无约束极小化方法求解等式约束非线性规划问题的Hestenes-Powell 增广拉格朗日函数作了进一步研究．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．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．     

关于用拉格朗日乘子法求解线性等式约束最小二乘问题

本文讨论用拉格朗日乘子法求解线性等式约束最小二乘问题(简称 LSE 问题)的优点.应用此法能细致地讨论约束条件与变量之间的关系,据此并可证明 LSE 问题与某一个无约束最小二乘问题的等价性.此外,尚可得到参数和拉格朗日乘子的协方差矩阵.最后给出一个数值稳定的解 LSE 问题的算法.     

无限多类别网络均衡问题中的收费计算

对于一个多类别的网络均衡问题,可以通过计算某个辅助问题的容量限制约束相应的乘子向量得到有效收费.本文通过计算拉格朗日函数的鞍点来计算乘子向量.借助于广义拉格朗日函数的稳定性和Uzawa算法非精确解的收敛性,得到鞍点序列的收敛性.其中离散化方法用于最小化广义拉格朗日函数的计算.     

两大部类扩大再生产中的广义拉格朗日乘子

在一个带有非负和不等式约束的优化问题有最优解的情形下,存在着广义拉格朗日乘子即资源的影子价格．本文探索给出马克思两大部类扩大再生产中的影子价格,为经典的马克思扩大再生产理论增添新的重要内容．首先使用“价值系数法”替代单纯形法,简便地求得了扩大再生产优化问题的最优解．然后运用库恩一塔克条件,确立了关于最优解与广义拉格朗日乘子的互补松弛条件的三个不等式组．进而利用这些不等式组和已知的最优解,简便地解出广义拉格朗日乘子,即两大部类扩大再生产中的影子价格．最后引用和借鉴《资本论》中的两个举例,对所获得的影子价格和目标函数最优值做了计算验证．     

初始点任意的解非线性不等式约束优化问题的结合共轭梯度参数的超记忆梯度广义投影算法

本文利用广义投影矩阵，对求解无约束规划的超记忆梯度算法中的参数给出一种新的取值范围以保证得到目标函数的超记忆梯度广义投影下降方向，并与处理任意初始点的方法技巧结合建立求解非线性不等式约束优化问题的一个初始点任意的超记忆梯度广义投影算法，在较弱条件下证明了算法的收敛性．同时给出结合FR，PR，HS共轭梯度参数的超记忆梯度广义投影算法，从而将经典的共轭梯度法推广用于求解约束规划问题．数值例子表明算法是有效的．     

单输入单输出大规模动力系统模型的简化方法

提出的简化单输入单输出大规模动力系统的一种新方法是系统在等式约束最小二乘法的一种推广.这种方法是一种投影方法,其投影依赖于奇异分解和Krylov子空间.通过平移算子,使得降阶模型与原模型的前r+i模准确地匹配,剩余的高阶模利用拉格朗日乘子法进行等式约束最小二乘的形式逼近原模.通过拉格朗日乘子法来求解具有约束条件的最小二乘问题,让推导出来的用于模型简化的投影变换矩阵更为简便.     

求解非线性等式和不等式约束优化问题的三项记忆梯度广义投影算法

利用广义投影矩阵，对求解无约束规划的三项记忆梯度算法中的参数给一条件，确定它们的取值范围，以保证得到目标函数的三项记忆梯度广义投影下降方向，建立了求解非线性等式和不等式约束优化问题的三项记忆梯度广义投影算法，并证明了算法的收敛性．同时给出了结合FR，PR，HS共轭梯度参数的三项记忆梯度广义投影算法，从而将经典的共轭梯度算法推广用于求解约束规划问题．数值例子表明算法是有效的．     

光测弹性理论中的耦联变分原理和广义耦联变分原理

在本文中，应用拉格朗日乘子法和高阶拉格朗日乘子法［１］，我们系统地导出了光测弹性理论中的耦联势能原理，耦联余能原理和具有二类和三类变量的广义耦联势能原理和广义相联余能原理．     

等式约束最小化的信整域乘子算法

本文通过使用信赖域乘子策略和引入不可微的势函数，讨论了[1]中被合理修正的双边投影拟牛顿方法，分析和叙述了算式约束最小化的信赖域乘子算法，并且证明了算法整体收敛性以及局部超越性收敛速率。     

分布欧拉方程与分片函数的表示

本文用分布(广义函数)的概念导出了刻画带内点约束的变分问题的解的分布欧拉方程,说明在这类问题中拉格朗日乘子法仍然是有效的。此外,利用基本解给出了分布欧拉方程的解的表示。进而给出了一元和多元广义变分样条函数的表示的一般方法。     

等式约束非线性控制系统的时程精细计算

针对等式约束非线性最优控制问题，通过一阶Taylor级数展开，得到线性化的动力学方程，进而在方程原变量的基础上，引入对偶向量（Lagrange乘子向量），将动力学方程从Lagrange体系引入到了Hamilton体系，在全状态下，从一个新的角度对等式约束非线性控制问题进行了描述，进一步基于时程精细积分理论，对其方程进行了有效的精细求解，并通过算例说明了中方法的有效性。     

爆炸荷载作用下地下复合结构动力分析

在地下抗爆结构的合理选型中，为了改善结构截面的受力状态，使截面各部位的材料强度得到充分的发挥，提出了复合结构的研究方法；采用微段隔离体分析的方法，给出了复合结构的平衡方程、约束方程和变形协调方程，利用广义功的概念直接引入物理意义明确的Lagrange乘子，应用变分方法证明了所构造的广义泛函的正确性，通过算例提出了复合结构截面合理的刚度匹配关系。     

约束Hamilton系统的Lie对称性及其在场论中的应用

研究了约束Hamilton系统的Lie对称性,得到了场论系统的守恒量.首先给出约束Hamilton系统的正则运动方程和固有约束方程;其次构建了约束Hamilton 系统的Lie对称性确定方程和结构方程;然后给出了约束Hamilton系统的Lie守恒定理和守恒量;最后研究了复标量场与Chern-Simons项耦合系统的Lie对称性和另外一个例子以说明此方法在场论中的应用.     

On the numerical determination of optimal inputs

The design of optimal inputs for linear and nonlinear system identification involves the maximization of a quadratic performance index subject to an input energy constraint. In the classical approach, a Lagrange multiplier is introduced whose value is an unknown constant. In recent papers, the Lagrange multiplier has been determined by plotting a curve of the Lagrange multiplier as a function of the critical interval length or a curve of input energy versus the interval length. A new approach is presented in this paper in which the Lagrange multiplier is introduced as a state variable and evaluated simultaneously with the optimal input. Numerical results are given for both a linear and a nonlinear dynamic system.     

具有等式与不等式约束条件拟可微优化的Lagrange乘子型最优性条件

讨论了不等式约束优化问题中拟微分形式下Ｆｒｉｔｚ Ｊｏｈｎ必要条件与 Ｃｌａｒｋｅ广义梯度形式下Ｆｒｉｔｚ Ｊｏｈｎ必要条件的关系．在较弱条件下给出了具有等式与不等式约束条件的两个Ｌａｇｒａｎｇｅ乘子形式的最优性必要条件,在这两个条件中等式约束函数的拟微分和Ｃｌａｒｋｅ广义梯度分别被使用。     

OPTIMAL PORTFOLIO ON TRACKING THE EXPECTED WEALTH PROCESS WITH LIQUIDITY CONSTRAINTS

In this article, the authors consider the optimal portfolio on tracking the expected wealth process with liquidity constraints. The constrained optimal portfolio is first formulated as minimizing the cumulate variance between the wealth process and the expected wealth process. Then, the dynamic programming methodology is applied to reduce the whole problem to solving the Hamilton-Jacobi-Bellman equation coupled with the liquidity constraint, and the method of Lagrange multiplier is applied to handle the constraint. Finally, a numerical method is proposed to solve the constrained HJB equation and the constrained optimal strategy. Especially, the explicit solution to this optimal problem is derived when there is no liquidity constraint.     

The Deterministic,Two-Product,Inventory System with Capacity Constraint

This paper considers the standard deterministic inventory system for two products with a capacity constraint and describes how to find the optimal policy amongst all policies which have fixed order quantities. This involves the idea of staggered initial orders and periodic policies, and includes the classical Lagrange multiplier technique and the equal order intervals method as special cases. It is shown that the usual Lagrange multiplier technique will never produce the optimal policy (in the class described above) except in the trivial case, when the capacity constraint is satisfied by the optimal unconstrained policy.     

Noether’s Theorem for Constrained Hamiltonian System Under Generalized Operators北大核心CSCD

Noether’s symmetry and conserved quantity of singular systems under generalized operators were studied. Firstly, the Lagrangian equation of singular systems under generalized operators was established, and the primary constraints on the system were derived. Then the Lagrangian multiplier was introduced to establish the constrained Hamilton equation and the compatibility condition under generalized operators. Secondly, based on the invariance of the Hamilton action under the infinitesimal transformation, Noether’s theorem for constrained Hamiltonian systems under generalized operators was established, and the symmetry and corresponding conserved quantity of the system were given. Under certain conditions, Noether’s conservation of constrained Hamiltonian systems under generalized operators can be reduced to Noether’s conservation of integer-order constrained Hamiltonian systems. Finally, an example illustrates the application of the results. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Optimal investment-reinsurance policy for an insurance company with VaR constraint

This paper investigates an investment-reinsurance problem for an insurance company that has a possibility to choose among different business activities, including reinsurance/new business and security investment. Our main objective is to find the optimal policy to minimize its probability of ruin. The main novelty of this paper is the introduction of a dynamic Value-at-Risk (VaR) constraint. This provides a way to control risk and to fulfill the requirement of regulators on market risk. This problem is formulated as an infinite horizontal stochastic control problem with a constrained control space. The dynamic programming technique is applied to derive the Hamilton-Jacobi-Bellman (HJB) equation and the Lagrange multiplier method is used to tackle the dynamic VaR constraint. Closed-form expressions for the minimal ruin probability as well as the optimal investment-reinsurance/new business policy are derived. It turns out that the risk exposure of the insurance company subject to the dynamic VaR constraint is always lower than otherwise. Finally, a numerical example is given to illustrate our results.     

Design of linear regulators with energy constraints

In the usual design of linear-quadratic optimal-control systems, the regulator performance is obtained for several different values of the constant Lagrange multiplier q. The Lagrange multiplier determines the amount of control energy expended. If the energy is to be constrained, then the value of q must be found such that the energy constraint is satisfied. In this paper a method is described for determining simultaneously the optimal trajectory and the value of q which satisfies the energy constraint.