从第一积分构造Lagrange函数的直接方法

提出力学系统Lagrange函数和第一积分之间存在一种新关联,在此基础上给出变分法逆问题的一种新的直接解法.证明系统Lagrange函数可以由带修正因子的第一积分构成,导出修正因子应满足的偏微分方程,运用此解法构建不同系统的Lagrange函数和函数族,并讨论新解法的特点.     

利用变量变换构造耗散系统Lagrange函数

提出变系数耗散系统变分法逆问题的一种间接解法.首先利用坐标和时间变量变换将系统运动微分方程化为自伴随方程,计算得到Lagrange函数后,再变换为原来变量以得到给定方程的Lagrange函数.给出两个例子以说明所得结果的应用.     

一阶系统自伴随条件和Lagrange函数的构造

重新研究一阶系统变分法逆问题,修正了某些文献关于一阶微分方程组自伴随条件的失误,导出了把一阶系统化为自伴随形式的变换矩阵所满足的方程,列出了构造自伴随一阶系统Lagrange函数的两种方法.举例说明所得结果的应用.     

从运动方程构造Lagrange函数的直接方法

根据对Lagrange函数的结构分析,提出直接从运动微分方程构造一维系统Lagrange函数新的一般方法和6种特殊方法.利用提出的方法导出若干运动微分方程的Lagrange函数.直接构造法证实一个系统具有多个不同而等效的Lagrange函数,甚至是Lagrange函数族.这种直接构造法也是构造Lagrange对称性并导出对应守恒量的一种途径.     

Lagrange 神经网络的稳定性分析

若重新定义与不等式约束相关的乘子为正定函数,则在构造Lagrange神经网络时,可直接使用处理等式约束的方法处理不等式约束,不需再用松驰变量将不等式约束转换为等式约束,减小了网络实现的复杂程度.利用Liapunov一阶近似原理,严格分析了这类Lagrange神经网络的局部稳定性;并采用LaSalle不变集原理,讨论其大范围稳定性.     

非保守非线性刚-热-弹耦合动力学的Lagrange方程

如何将Lagrange方程应用于弹性动力学,一直是国内外学术界关注的理论和应用研究课题.在这类问题获得基本解决之后,Lagrange方程应用于耦合动力学的理论难题又摆在我们的面前.本文采用Lagrange Hamilton体系,成功地将Lagrange方程应用于非保守非线性刚 热 弹耦合动力学.进而应用非保守非线性刚 热 弹耦合动力学的Lagrange方程推导出非保守非线性刚 热 弹耦合动力学的控制方程.讨论了应用耦合动力学的Lagrange方程解决实际工程技术问题的途径.     

基于Lagrange插值多项式拟合的力学系统的变分积分子

变分积分子是通过直接离散变分原理得到的一类特殊的动力学系统的数值差分格式,较之传统差分格式呈现出明显的计算优越性.由离散Euler-Lagrange方程的形式可知,变分积分子的构造过程最终归结为计算离散Lagrange函数的偏导数,其中离散Lagrange函数是Lagrange函数在单个时间步长的积分,通常由经典求积公式近似得到.根据离散Lagrange函数的积分表达式,解析计算其偏导数会随之衍生一个新的且与连续Euler-Lagrange方程密切关联的积分,因此,构造变分积分子就可以不再以通过经典求积公式得到的具体形式的离散Lagrange函数为前提,而是可以直接基于一组离散结点近似新衍生的积分.在这些离散结点处,如果进一步让系统的拟合轨迹严格满足Euler-Lagrange方程,即运动方程,那么新的积分自动为零,相应地,计算离散Lagrange函数的偏导数就简化为计算连续Lagrange函数关于速度变量的偏导数.这种新的构造方式同时结合了连续和离散的Euler-Lagrange方程,不仅让最终得到的差分格式仍然继承了变分积分子特有的优越计算性能,而且在同阶精度的情况下具有更小的局部误差.     

非定常完整约束系统的线性映射方法

通过一阶线性映射可以从非定常完整约束系统的位形空间映射出一个时空Π,并诱导出时空Π上的附加几何结构(度规和联络),由此可以写出约束系统在时空Π中的运动方程.当一阶线性映射不可积时,时空Π是一个Riemann Cartan空间;当一阶线性映射可积时,时空Π将退化为一个Riemann空间,且此时由这种线性映射方法得到的时空Π中的运动方程等价于用广义坐标表示的约束系统的Lagrange方程.     

关于一类Lagrange函数族的存在条件

研究力学系统的Lagrange函数族,根据从第一积分直接构造Lagrange函数的方法,提出一类Lagrange函数族的结构,导出这类Lagrange函数族的存在条件和直接构造方法.举例说明所得结果的应用.     

一类Lagrange坐标系下的ENO有限体积格式

本文首先从积分形式的二维Lagrange流体力学方程组出发,使用ENO高阶插值多项式,推广了四边形结构网格下的一阶有限体积格式,构造得到了一类结构网格下的高精度有限体积格式.该格式针对单介质问题具有良好的计算效果,同时在处理多介质问题时,不会产生物质界面附近强烈的震荡.结合有效的守恒重映方法,用ALE方法进行数值模拟,得到了预期的效果.     

An improved stepsize of the subgradient algorithm for solving the lagrangian relaxation problem

The Lagrangian relaxation approach has been successfully applied to many large-scale mathematical programming problems. The Lagrangian relaxation problem itself is a non-differentiable optimization problem. One of the methods for solving such problem is the subgradient algorithm. In this paper, we propose an improved stepsize of the subgradient algorithm for solving the Lagrangian relaxation problem. Our version of the algorithm may significantly improve the rate of convergence of subgradient algorithm when applied to the solution of Lagrangian relaxation problem. An illustrative numerical example is also given.     

A hybrid differential evolution augmented Lagrangian method for constrained numerical and engineering optimization

We present a new hybrid method for solving constrained numerical and engineering optimization problems in this paper. The proposed hybrid method takes advantage of the differential evolution (DE) ability to find global optimum in problems with complex design spaces while directly enforcing feasibility of constraints using a modified augmented Lagrangian multiplier method. The basic steps of the proposed method are comprised of an outer iteration, in which the Lagrangian multipliers and various penalty parameters are updated using a first-order update scheme, and an inner iteration, in which a nonlinear optimization of the modified augmented Lagrangian function with simple bound constraints is implemented by a modified differential evolution algorithm. Experimental results based on several well-known constrained numerical and engineering optimization problems demonstrate that the proposed method shows better performance in comparison to the state-of-the-art algorithms.     

等待时间有限的job-shop调度

对一类产品加工相邻步骤之间具有等待时间约束的ｊｏｂ－ｓｈｏｐ调度问题进行了建模，并采用Ｌａｇｒａｎｇｉａｎ松弛法结合动态规划对这类问题进行求解，提出了一种基于集束式搜索的启发式方法，用于从对偶问题的解构作原问题的可行解．最后给出了仿真计算结果．     

A heuristic algorithm based on Lagrangian relaxation for the closest string problem

The closest string problem that arises in both computational biology and coding theory is to find a string minimizing the maximum Hamming distance from a given set of strings. This study proposes an efficient heuristic algorithm for this NP-hard problem. The key idea is to apply the Lagrangian relaxation technique to the problem formulated as a mixed-integer programming problem. This enables us to decompose the problem into trivial subproblems corresponding to each position of the strings. Furthermore, a feasible solution can be easily obtained from a solution of the relaxation. Based on this, a heuristic algorithm is constructed by combining a Lagrangian multiplier adjustment procedure and a tabu search. Computational experiments will show that the proposed algorithm can find good approximate solutions very fast.     

On the geometry of symplectic pencils arising from discrete-time matrix equations

In this paper general symplectic matrix pencils are considered disregarding the particular matrix equations from which they arise. A parameterization of the Lagrangian deflating subspaces is given with the only assumption of regularity of the matrix pencil.     

A Lagrangian relaxation technique for the general assembly line balancing problem

This paper addresses the problem of balancing assembly or fabrication lines. In order to achieve a given production rate or to optimize the use of workstations, one has to tackle the problem of balancing the production lines. It is well known that this problem belongs to the class of NP-hard problems. In this paper the polyhedron of the feasible solutions of the assembly line balancing problem is first studied. Then a Lagrangian relaxation algorithm that incorporates the set of cycle constraints in the objective function is proposed. These constraints are the complicating restrictions in the model. The relaxed problem has the interesting property that its linear programming relaxation always has integer optimal solutions. The subgradient algorithm is then used to maximize the Lagrangian dual. A heuristic is also used to find primal feasible solutions for the original line balancing integer program. These two bounds are then used to reduce the size of the branch-and-bound tree.     

Pinning Synchronization in Networked Lagrangian Systems

This paper investigates the synchronization problem of networked Lagrangian systems based on pinning control framework on complex networks. We propose a pinning algorithm to guarantee the controlled synchronization of networked identical Lagrangian systems by applying local linear feedback injections to a small fraction of nodes. We also present some simple yet generic criteria on pinning synchronization for such an algorithm over undirected connected graphs, where all the agents are regulated to follow a synchronization state. Furthermore, the pinning controllability in networked Lagrangian systems is also discussed. Compared with some existing works on networked Lagrangian systems, the distinctive advantages of the proposed pinning algorithm include: (i) independence on the knowledge of system models; (ii) explicit consideration of agent's intrinsic complex dynamics; and (iii) simplicity of implement procedure in practice. Subsequently, the results are illustrated by a typical Lagrangian network composing of eight two‐link revolute manipulators. Numerical simulations with different kinds of pinning schemes are finally given to demonstrate the effectiveness of the proposed control methodology.     

A hybrid Lagrangian genetic algorithm for the prize collecting Steiner tree problem

We consider the version of prize collecting Steiner tree problem (PCSTP) where each node of a given weighted graph is associated with a prize and where the objective is to find a minimum weight tree spanning a subset of nodes and collecting a total prize not less that a given quota Q.$Q.$ We present a lower bound and a genetic algorithm for the PCSTP. The lower bound is based on a Lagrangian decomposition of a minimum spanning tree formulation of the problem. The volume algorithm is used to solve the Lagrangian dual. The genetic algorithm incorporates several enhancements. In particular, it fully exploits both primal and dual information produced by Lagrangian decomposition. The proposed lower and upper bounds are assessed through computational experiments on randomly generated instances with up to 500 nodes and 5000 edges. For these instances, the proposed lower and upper bounds exhibit consistently a tight gap: in 76% of the cases the gap is strictly less than 2%.