线性流形上复对称矩阵的最小二乘问题

本文主要讨论了线性流形上复对称矩阵的最小二乘问题。在推导出所给线性流形中任意矩阵的显式表达的基础上,利用奇异值分解和Frobenius范数的酉不变性得到了该最小二乘问题通解的一般表达式。此外,文章还考虑了任一给定矩阵对此最小二乘问题解集合的最佳逼近问题,证明了该最佳逼近问题存在唯一解,并利用酉矩阵的性质得到了最佳逼近解的表达式。     

子矩阵约束下矩阵方程ATXA=B的实矩阵解及其最佳逼近

本文讨论了子矩阵约束下一类矩阵方程的实矩阵解问题。基于矩阵的奇异值分解和广义奇异值分解方法,给出了该问题有解的充要条件和解的一般表达式。并证明了对任一给定的实矩阵,在上述解集合中必存在唯一的最佳逼近解,给出了最佳逼近解的形式。     

多目标全系数模糊线性规划

在模糊线性规划理论的基础上,对同时含有等式约束和不等式约束的多目标全系数模糊线性规划的最优解进行研究。首先定义了目标协调度的概念,然后以目标协调度最大化为最优性条件,提出多目标全系数模糊线性规划的模糊最优解的定义,最后给出一种可行的求解方法。     

双中心矩阵反问题及其在电网络理论中的应用

本文研究双中心矩阵反问题。建立了双中心矩阵反问题的最小二乘解,得到了解的具体表达式。讨论了用双中心矩阵反问题的解构造给定矩阵的最佳逼近问题,给出该问题有解的充分必要条件和解的表达式。设计了相应的算法并给出了其在电网络理论中的应用。     

基于随机平均的非线性随机最优控制

首先建立非线性随机系统的最优控制问题,并介绍通过随机平均法导出平均系统、再由随机动态规划原理确定控制律的平均系统的非线性随机最优控制方法。然后,对于非线性随机系统的动态规划方程,提出应用随机平均法简化该方程、从而得到最优平均控制律的方法,并证明该最优平均控制律等价于平均系统的最优控制律。最后用一个例子说明方法及等价性,并指出在一定条件下,最优平均控制律将是动态规划方程的精确解。     

哈明距离下极大不一致支撑树的部分逆问题

给定一个简单无向赋权图和其中的一个森林,极大不一致支撑树的部分逆问题研究如何尽可能少地改变图中各边的权,使得在新的权值下存在一个极大不一致支撑树包含该森林.在赋权哈明距离下,得到了该问题的一些性质,并且给出了求解该问题的多项式时间算法.     

微生物连续培养非线性动力系统的性质及最优性条件

本文针对微生物连续发酵生产1,3-丙二醇非线性动力系统,以产物1,3-丙二醇的生产强度最大为目标泛函,建立最优控制模型。用不可微优化理论与方法证明了模型最优解的存在性与一阶最优性条件。论述了最优性函数与一阶最优性条件的等价性。     

有高阶矩约束的最优投资组合模型及近似线性规划解法

本文讨论有包括偏度和峰度在内的高阶矩约束的最优投资组合模型。证明了最优投资组合决策的存在性并导出解析解的隐式表达式,然后利用线性逼近的方法得到近似解,并给出了具体算例,最后分析了模型中的权重参数对最优目标的影响。     

Time-variant lot sizing models for the warehouse scheduling problem

We consider the problem of scheduling the delivery of n products into a warehouse with limited space under the assumptions of continuous demands at constant rates, infinite horizon, and no backorders. The delivery schedule is described by a cyclic schedule with time-varying lot sizes. The order frequencies and the order sequence are assumed to be given. We formulate a linear program that determines delivery times relative to the cycle length to minimize the relative maximum space used and show that the optimal solution is characterized by filling the warehouse at each order. We bound the optimal solution by using a worst-case analysis and give conditions under which the linear program has the same optimal solution as a quadratic program that minimizes the holding cost. Under general conditions, we derive a bound on the cost penalty that results when using the optimal solution of the linear program as a solution to the quadratic program. Finally, we complete a solution to the nonlinear lot-sizing model by determining the best cycle length corresponding to the solution to the linear program and present a bound on a quality of this solution.     

线性流形上广义次对称矩阵的最佳逼近

讨论了线性流形上广义次对称矩阵的最小二乘解,得到了解的一般表达式,对于任意给定的实对称矩阵A,在最小二乘解集中得到了A的最佳逼近解。     

一类线性矩阵方程的最佳逼近解

本文利用投影定理、广义奇异值分解和标准相关分解技巧给出了一种求矩阵方程AXB=C的最小二乘反对称解的方法,得到了通解表达式。进而利用此表达式,导出了通解集做为一个矩阵集与任意给定矩阵的最小距离元素。     

招聘公务员问题的优化模型与评述

本文针对2004年高教社杯全国大学生数学建模竞赛的D题"招聘公务员"问题的评卷情况,首先概括地介绍了这个问题的背景、评卷要点、问题的解决方法和答卷中存在的问题。最后给出了解决这个问题的一个优化模型及求解结果。     

The optimal control of fracture and stress parameters in a piezoceramic halfspace with cracks

An approach to the determination of the optimal control of fracture and strength parameters in a piezoceramic halfspace with cracks under antiplane deformation conditions is proposed. The distribution over certain part of the halfspace boundary, harmonically changing with time under the influence of axial forces or electric charges is analyzed as a control interaction. The solution of the inverse problem in fracture mechanics is obtained from the solution of the corresponding direct boundary value problem; in this case, the optimization problem is reduced to the momentum problem. The solution of the direct electroelastic problem using the method of the boundary integral equations is obtained. Various control functions permitting to realize the optimal process of control, i.e. the minimal energetic expenses, are given.     

An Explicit Solution for Generalized Ridge Regression

The general form of ridge regression proposed by Hoerl and Kennard is examined in the context of the iterative procedure they suggest for obtaining optimal estimators. It is shown that a non-iterative, closed form solution is available for this procedure. The solution is found to depend upon certain convergence/divergence conditions which relate to the ordinary least) squares estimators. Numerical examples are given.     

Parametric optimal control of uncertain systems under an optimistic value criterion

It is well known that the optimal control of a linear quadratic model is characterized by the solution of a Riccati differential equation. In many cases, the corresponding Riccati differential equation cannot be solved exactly such that the optimal feedback control may be a complex time-oriented function. In this article, a parametric optimal control problem of an uncertain linear quadratic model under an optimistic value criterion is considered for simplifying the expression of optimal control. Based on the equation of optimality for the uncertain optimal control problem, an approximation method is presented to solve it. As an application, a two-spool turbofan engine optimal control problem is given to show the utility of the proposed model and the efficiency of the presented approximation method.     

The problem of the selection of an a posteriori error indicator based on smoothening techniques

This paper addresses the problem of assessing the quality of an a posteriori error estimate of a finite element solution. An error estimate based on local L2-projections is analysed in the case of translation-invariant meshes. It is shown that for general meshes this technique does not lead to an asymptotically exact estimator. The problem is analysed in detail in the one-dimensional setting. It is shown that an asymptotically exact estimator is not the optimal one when the solution is not sufficiently smooth. An optimal estimator for adaptively constructed meshes is given. Finally, a general mathematical framework for the quality assessment of estimators is introduced.     

矩阵方程AX-BY=Z的最小二乘中心对称解及其最佳逼近

本文研究矩阵方程AX-BY=Z的最小二乘中心对称解,给出了AX-BY=Z的最小二乘中心对称解的表达式,导出了AX-BY=Z有中心对称解的条件。讨论了在AX-BY=Z的最小二乘中心对称解集合中求与给定矩阵最佳逼近的解,并将所得结果应用于研究一类中心对称矩阵的广义特征值反问题。     

A hybrid ‘dynamic programming/depth-first search’ algorithm, with an application to redundancy allocation

A hybrid 'dynamic programming/depth-first search' algorithm has been developed to solve non-linear integer programming problems arising in the reliability optimization of redundancy allocation. Initially, the technique solves the knapsack relaxation of the original mathematical programming problem using dynamic programming. Then, all solutions in some range of the relaxation problem are obtained via an enumerative depth-first search technique. The solutions are ranked and the optimal solution is given by the best one that satisfies the remaining constraints of the given problem. Computational complexity of the algorithm is also discussed. The salient features of our hybrid algorithm are its simplicity and ease of programming. Our algorithm also has an advantage over the traditional Lagrangian and surrogate dual approaches. It does not have to deal with the issue of 'duality gap' as in classical dual approaches, which is responsible for the failure to identify optimal solutions to the primal integer optimization problems. Of most importance, it guarantees to succeed in identifying an optimal solution.     

矩阵方程AXB=C的中心对称最小二乘解及其最佳逼近的迭代算法

本文建立了求矩阵方程AXB=C的中心对称最小二乘解的迭代算法。在不考虑舍入误差时,对任意给定的初始中心对称矩阵,该算法能够在有限步迭代后得到此方程的中心对称最小二乘解。当选取特殊的初始矩阵时,可得到极小范数中心对称最小二乘解。另外,在上述解集合中也可得到给定矩阵的最佳逼近矩阵的表达式。