基于Newmark法敏度计算的刚架结构动力优化

提出一种有效的求解结构最小质量设计,同时满足动位移和动应力约束的二阶优化设计方法。在有限元法和纽马克法基础上导出一种高效的动应力、动位移对设计变量一阶导数和二阶导数的算法。建立含时间参数,以结构质量最小化为目标,同时满足动位移、动应力和设计变量约束的优化数学模型,通过积分型内点罚函数将含时间参数的不等式约束优化问题转变为一系列不含时间参数的无约束优化问题。利用动位移、动应力对设计变量一阶导数和二阶导数的信息计算内点罚函数的梯度和海森矩阵,利用梯度和海森矩阵构造求解优化设计问题高效有效的二阶优化算法。算例结果表明该文的优化设计方法能获得刚架结构的局部最优设计,优化的效率高于增广拉格朗日乘子法。     

曲面间最短距离的确定

寻求解决工程设计中曲面最短距离判定与确定的合理方法。以两个曲面上未知点坐标和乘子未知参量为设计变量,将带有拉格朗日乘子的距离函数偏导数与约束函数的平方和作为目标函数,建立了精确判定和确定两个曲面之间最短距离的无约束优化方法。基于对曲面最短距离分析,给出解决问题的方法与计算程序。通过两个交叉锥面的计算实例,表明研究方法具有有效性和实用性。     

基于接触协同作用结构非线性屈曲分析

工程结构中结构或构件之间存在接触摩擦现象是较为普遍的,接触协同作用使得结构非线性屈曲问题的数值求解变得困难。该文从这一实际问题出发,采用子结构凝聚自由度的有限元方法得到基于接触界面问题的非线性有限元平衡方程,通过增广拉格朗日乘子法对接触问题的约束条件进行处理,并与柱面弧长法结合,对非线性屈曲问题进行全过程跟踪求解。算例结果表明接触协同作用对结构的非线性屈曲过程影响明显,有必要在实际工程结构设计中考虑接触协同作用的影响。     

非完整移动机器人路径跟踪的快速非线性模型预测控制方法

本文用快速非线性模型预测控制方法进行了非完整移动机器人路径跟踪问题的研究.采用虚拟目标跟踪法建立非完整移动机器人路径跟踪问题的非线性系统,转化为以Bolza形式的最优控制问题模型.通过一种障碍函数法处理不等式约束将其整合至性能指标函数中,基于拉格朗日乘数法建立最优化必要条件的线性方程组,以连续/广义极小残余算法(C/G...     

机构系统关节约束反力分析

为了研究机构系统约束反力的计算方法,在考虑第一类拉格朗日方程中拉格朗日乘子与约束反力关系的基础上,明确了拉格朗日乘子项是广义约束力中各个约束所占的权重。通过拉格朗日增广法计算了机构系统中关节的反力,详述了每个关节对应关节反力的计算流程和筛选步骤。为了证明计算方法的正确性,利用牛顿欧拉法对所选研究对象的关节反力再次求解,并通过ADAMS仿真软件进行动力学仿真,将三种结果进行对比,验证了利用拉格朗日增广法计算机构对应关节反力时所用筛选方法的正确性。     

一个新的单参数填充函数算法

本文研究连续全局最优化问题的确定性求解方法.构造了一个单参数填充函数并证明了该填充函数的性质.该填充函数算法由极小化阶段和填充阶段两个阶段构成.其中极小化阶段利用局部优化方法获得填充函数的局部极小点,对填充函数的无约束极小化使得算法离开原目标函数的任何局部极小点.填充阶段依据原目标函数的局部极小点构造填充函数.极小化阶段和填充阶段交替重复实施直到终止准则满足.最后,给出了填充函数算法的数值结果.     

桁架动力学形状优化的统一设计变量方法

研究了具有多种约束 ,特别是动力学约束 (频率约束 )作用下的平面桁架形状优化问题。提出一种将两类不同性质的设计变量 (尺寸变量、节点几何坐标变量 )变换为统一形式的无量纲设计变量的方法 ,解决了不同性质变量耦合引起的收敛困难问题 ,并拓展了设计空间。联合运用内点罚函数法、DFP法 (即变尺度法 )和一维搜索技术 (二次插值法 ) ,将约束优化问题转化为无约束序列优化问题 ,得到了满意的优化结果。算例表明本文方法对桁架形状优化的有效性 ,并显示了算法的简洁性和工程设计实用性     

约束全局优化问题的一个单参数填充函数方法

类似于无约束全局优化问题,本文给出了求解约束全局优化问题的一个填充函数方法,首先给出了约束全局优化问题的填充函数定义,在此定义的基础上提出了一个单参数填允函数.讨论了该函数的性质,并设计了一个填充函数算法,数值计算结果显示该算法是有效的和可行的.     

服务水平约束下的提前期压缩问题的研究

在一定服务水平约束条件下,提出了新的具有分段可微特征的提前期压缩成本函数,构建了两阶段(Q,r)库存模型,并采用优化理论将模型需求解的问题分解为两个新的问题:无约束问题和等式约束问题.最后利用成本函数分段可微的特性结合数学分析的知识得到无约束问题的解,同时对于等式约束问题则利用了拉格朗日乘子法求得最优解.理论分析及算例表明,可以通过合理的确定模型申的订货量Q、订货点r、提前期加速因子τ等参数值使得在服务水平显著提高的前提下成本最小化.     

基于拉格朗日乘子法的空间圆弧拟合优化方法

针对传统的空间圆弧拟合方法鲁棒性低、拟合精度不高等问题,提出了一种鲁棒性较强的空间圆弧拟合优化方法。首先,以拉格朗日乘子法为基础,基于平面条件约束建立目标函数,从而得出空间圆弧拟合方程;其次,采用RANSAC（random sample consensus,随机抽样一致）算法剔除错误跟踪点,将RANSAC算法的高稳定性应用到空间圆弧拟合的点云优化中,进而提高拟合精度。最后,通过实验分析验证了所提空间圆弧拟合优化方法的可行性,并与传统拟合方法进行比较,分析所提方法的拟合精度。实验结果表明：普通圆弧点云拟合的相对精度在0.003左右,复杂圆弧点云拟合的相对精度在0.01左右;相较于传统拟合方法,所提方法有效解决了拟合精度低及鲁棒性差等问题。研究结果表明提出的空间圆弧拟合优化方法一方面可运用拉格朗日乘子法增强鲁棒性,另一方面可通过采用RANSAC方法剔除错误点以提高拟合精度,具有广泛的工程实际应用价值。     

An alternating optimization approach for mixed discrete non-linear programming

This article contributes to the development of the field of alternating optimization (AO) and general mixed discrete non-linear programming (MDNLP) by introducing a new decomposition algorithm (AO-MDNLP) based on the augmented Lagrangian multipliers method. In the proposed algorithm, an iterative solution strategy is proposed by transforming the constrained MDNLP problem into two unconstrained components or units; one solving for the discrete variables, and another for the continuous ones. Each unit focuses on minimizing a different set of variables while the other type is frozen. During optimizing each unit, the penalty parameters and multipliers are consecutively updated until the solution moves towards the feasible region. The two units take turns in evolving independently for a small number of cycles. The validity, robustness and effectiveness of the proposed algorithm are exemplified through some well known benchmark mixed discrete optimization problems.     

Second‐order optimal control algorithm for complex systems

The solution to large‐scale optimal control problems, characterized by complex dynamics and extended time periods, is often computationally demanding. We present a solution algorithm with favourable local convergence properties as a way to reduce simulation times. This method is based on using a trapezoidal direct collocation to convert the differential equations into algebraic constraints. The resulting constrained minimization problem is then solved with an augmented Lagrangian formulation to accommodate both equality and inequality constraints. In contrast to the prevalent optimal control software implementations, we calculate analytical first and second derivatives. We then apply a generalized Newton method to the augmented Lagrangian formulation, solving for all unknowns simultaneously. The computational costs of the Hessian formation and matrix solution remain manageable as the system size increases due to the sparsity of all tensor quantities. Likewise, the total iterations for convergence scale well due to the local quadratic convergence of the generalized Newton method. We demonstrate the method with an inverted pendulum problem and a neuromuscular control problem with complex dynamics and 18 forcing functions. The optimal control solutions are successfully found. In both examples, we obtain quadratic convergence rates in the neighbourhood of the solution. Copyright © 2002 John Wiley & Sons, Ltd.     

Active constrained truncated Newton method for simple-bound optical tomography

In the past, nonlinear unconstrained optimization of the optical imaging problem has focused on Newton-Raphson techniques. Besides requiring expensive computation of the Jacobian, the unconstrained minimization with Tikhonov regularization can pose significant storage problems for large-scale reconstructions, involving a large number of unknowns necessary for realization of optical imaging. We formulate the inverse optical imaging problem as both simple-bound constrained and unconstrained minimization problems in order to illustrate the reduction in computational time and storage associated with constrained image reconstructions. The forward simulator of excitation and generated fluorescence, consisting of the Galerkin finite-element formulation, is used in an inverse algorithm to find the spatial distribution of absorption and lifetime that minimizes the difference between predicted and synthetic frequency-domain measurements. The inverse approach employs the truncated Newton method with trust region and a modification of automatic reverse differentiation to speed the computation of the optimization problem. The reconstruction results confirm that the physically based, constrained minimization with efficient optimization schemes may offer a more logical approach to the large-scale optical imaging problem than unconstrained minimization with regularization.     

Three-dimensional unconstrained and constrained image-reconstruction techniques applied to fluorescence, frequency-domain photon migration

The development of near-infrared (NIR) optical imaging for biomedical optical imaging is hampered by the computational intensiveness of large-scale three-dimensional (3-D) image reconstruction and the potential lack of endogenous contrast for detection of relevant tissue features. In this contribution the inverse optical imaging problem is formulated in three dimensions in a noncompressive geometry as a simple-bound constrained minimization problem in order to recover the interior fluorescence properties of exogenous contrast agent from frequency-domain photon migration measurements at the boundary. The solution of the forward optical diffusion problem for the frustum shape containing fluorescence inclusions of 10:1 contrast is accomplished by use of the Galerkin finite-element formulation. The inverse approach employs the truncated Newton method with trust region and a modification of automatic reverse differentiation to speed the computation of the optimization problem. The image-reconstruction results confirm that the constrained minimization may offer a more logical approach for the 3-D optical imaging problem than unconstrained optimization.     

球约束凸二次规划的一个算法

针对球约束凸二次规划问题，利用Lagrange对偶将其转化为无约束优化问题，然后运用单纯形法对其求解，获得原问题的最优解。最后，对文中给出的算法给出了论证。     

Adaptive modification of objective function for lagrange multiplier method in constrained optimization problems

Abstract

This paper proposes an adaptive modification method to transform the objective function with a stationary point to an objective function with a minima point, such that search methods can be used to find the stationary point. The stationary point can be a saddle point in addition to a minima or a maxima. Therefore, this method can be used to transform a constrained optimization by applying Lagrange multipliers to an unconstrained optimization problem. A quadratic term, ½(X — X_N ) ^{T D} (X—X_N ), is added to the original function such that the modified function is a minima at the Newton point X_N of the original function, where D is a diagonal matrix to make the modified Hessian matrix H_O + D positive definite, and H_O is the original Hessian matrix at the initial point X_O .     

通过广义D-间隙函数求解变分不等式问题的全局收敛性的推广

变分不等式问题(VIP)可以通过D-间隙函数转化为一个无约束最优化问题。最近，Peng提出了一种混合型Newton方法来极小化D—间隙函数。本文对Pens的算法中作了适当的修改，建立了一个更强的全局收敛性定理，所得的结果推广了相应文献中的结果。     

An augmented Lagrangian approach to essential boundary conditions in meshless methods

The problem of boundary conditions enforcement in meshless methods has been solved in the literature by several approaches. In the present paper, the moving least‐squares (MLS) approximation is introduced in the total potential energy functional for the elastic solid problem and an augmented Lagrangian term is added to satisfy essential boundary conditions. The method can be easily extended to any kind of constraint for the approximation variables. The solution is found by iterating alternatively on approximation variables and on Lagrange multipliers. The advantages of the proposed formulation are: (a) the ability to deal with the same approach with any constraint type; (b) the number of the variables is not increased by the Lagrange multipliers; (c) the Hessian of the functional w.r.t. the approximation variables is banded, well conditioned and strictly positive definite and (d) the cost of the augmented Lagrangian iteration is a very small fraction of the global computing time. Therefore, the augmented Lagrangian element‐free (ALEF) approach represents an attractive and very efficient numerical tool, not only for the boundary conditions enforcement, but also for the solution of interface and non‐linear problems. Copyright © 2001 John Wiley & Sons, Ltd.     

On the relationship between a continuous update method of multipliers and the generalized reduced gradient method

The method of multipliers^1–3 (MOM) is a transformation technique which has enjoyed considerable popularity in recent years. The algorithmic philosophy is similar to conventional penalty function methods in that a constrained nonlinear programming problem is transformed into a sequence of unconstrained problems. In the standard MOM approach, the multipliers are updated after each unconstrained search. In this paper we investigate methods which involve continuous updating of the penalty parameters and design variables. We demonstrate that this continuous updating scheme is equivalent to the generalized reduced gradient method^4,5 applied to a certain dual problem. Computational results are given which suggest that the continuous updating MOM is not as efficient as one might reasonably hope.