最小二乘问题及其研究

本文从理论上讨论了线性方程中最小二乘解的存在性及最小范数最小二乘解的唯一性，并给出求最小二乘解及最小范数最小二乘解的公式方法。     

关于核反应堆系统的最小范数控制

讨论核反应堆系统散射裂变截面的最小范数控制问题。在一定条件下证明了最小范数散射裂变截面控制变量的存在性和唯一性，并给出其相应的优化条件。     

关于最小二乘QR分解算法(LSQR)的一个注记

本文从最小多项式出发,通过寻找包含奇异线性系统Ax=b最小范数解的一个解空间,获得了一个更简单的求解广义逆的计算公式.并从理论上对最小二乘QR分解算法（LSQR）收敛性进行了简单分析,分析表明LSQR的收敛性与矩阵A的非零奇异值密切相关,并用A的非零奇异值以及所寻找到的最小范数解空间将最小范数解线性表出.     

F_q[x]上的素数定理和Dirichlet定理及最小范数不可约多项式

讨论了F_q[x]上的zeta函数和L函数的解析性质,并在不假定黎曼猜想的情况下,导出了F_q[x]上的多项式环及其算术级数中不可约多项式的分布.然后,通过一系列的技术性处理,给出了算术级数中不可约多项式的最小范数的估计.成功地把素数定理及Dirichlet定理推广到了F_q[x]中,最重要的是,对应于最小素数问题,得到的最小范数的估计值本质上要比有理整数环上假定黎曼猜想情况下所推得的结果还好.     

Fq[x]上的素数定理和Dirichlet定理及最小范数不可约多项式

讨论了Fq[x]上的zeta函数和L函数的解析性质,并在不假定黎曼猜想的情况下,导出了Fq[x]上的多项式环及其算术级数中不可约多项式的分布.然后,通过一系列的技术性处理,给出了算术级数中不可约多项式的最小范数的估计.成功地把素数定理及Dirichlet定理推广到了Fq[x]中,最重要的是,对应于最小素数问题,得到的最小范数的估计值本质上要比有理整数环上假定黎曼猜想情况下所推得的结果还好.     

一类弹性振动系统的控制问题

本文讨论具结构阻尼系数的细长体飞行器的弹性振动方程支配系统的最优控制问题 .本文将结构阻尼系数作为控制变量 ,以“范数最小”来衡量其最优性 .证明了弹性振动系统存在唯一的最优控制元     

两类矩阵方程的极小范数最小二乘三对角Hermite解

正1引言矩阵方程广泛应用于诸多领域,例如:控制理论[1],系统稳定性分析[2]等.对矩阵方程的研究虽然已取得一系列重要成果[3-9],但仍然是数值代数领域中热门的课题之一.此外,由于三对角矩阵在诸多学科领域中的广泛应用,使得三对角矩阵倍受人们的关注.文献[10]利用Moore-Penrose广义逆及Kronecker积,给出四元数矩阵方程AXB=C的三对角Hermite极小范数最小二乘解和三对角双Hermite极小范数最小二乘解;文献[11]利用矩阵的实表示结构,给出四元数矩阵方程AXB=C的三对角Hermite极小范数最     

混合分数O-U过程的最小范数估计

本文研究混合分数O-U过程的最小范数估计问题.利用分数布朗运动驱动的随机微分方程偏差不等式,获得了混合分数O-U过程漂移参数的最小范数估计、相合性及渐近分布.     

四元数矩阵方程(AXB,CXD)=(E,F)的最小范数最小二乘Hermitian解

提出了研究四元数矩阵方程(AXB, CXD)=(E, F)的最小范数最小二乘Hermitian解的一个有效方法.首先应用四元数矩阵的实表示矩阵以及实表示矩阵的特殊结构,把四元数矩阵方程转化为相应的实矩阵方程,然后求出四元数矩阵方程(AXB, CXD)=(E, F)的最小二乘Hermitian解集,进而得到其最小范数最小二乘Hermitian解.所得到的结果只涉及实矩阵,相应的算法只涉及实运算,因此非常有效.最后的两个数值例子也说明了这一点.     

一种七自由度冗余机械臂的逆运动学优化算法

在建立一种常见的七自由度冗余机械臂D-H模型的基础上,结合固定关节角法和加权最小范数法,提出了一种基于二次计算的逆运动学优化算法.基于加权最小范数法具有回避关节极限以得到优化解的优势,该算法一方面在牺牲一定时间复杂度的条件下进一步提高了逆运动学求解的精度,另一方面基于迭代算法解决了雅克比伪逆不存在时加权最小范数法无法求解的问题.通过仿真结果可以看出,基于加权最小范数法的二次计算法在对期望轨迹的跟踪精度上有了很大提高,并能够较好地解决雅克比伪逆不存在时的求解问题.     

Minimum norm solutions for cooperative games

A solution f for cooperative games is a minimum norm solution, if the space of games has a norm such that f(v) minimizes the distance (induced by the norm) between the game v and the set of additive games. We show that each linear solution having the inessential game property is a minimum norm solution. Conversely, if the space of games has a norm, then the minimum norm solution w.r.t. this norm is linear and has the inessential game property. Both claims remain valid also if solutions are required to be efficient. A minimum norm solution, the least square solution, is given an axiomatic characterization.      

Existence of singular extremals and singular functionals in reachable spaces

Given a linear, infinite dimensional control system with point target and "full" control we show that singular extremals for the minimum norm problem exist except in certain exceptional cases ("singular" means "not satisfying Pontryagin's maximum principle"). Existence of singular extremals implies existence of certain functionals (also called singular) in the space of reachable states. Received March 5, 2001; accepted April 10, 2001.     

A class of functionals on a banach space for which strong and weak local minima do coincide ∗

We present a class of functionals on a real Banach space for which every local minimum point with respect to the norm topology is so also with respect to the weak topology     

Convergence of minimizing sequences

A functional f defined on a closed convex subset C of a normed space is to be minimized. It is known that if f is strictly convex and C is compact, then any minimizing sequence converges in norm to a unique minimum. A characterization is given herein for the norm convergence of any minimizing sequence when C is weakly compact and f is strictly quasi-convex, a more general result than those which are already known.     

局部凸空间中集值映射的极小不动点定理及其应用

利用局部凸空间中Fan-Kakutani不动点定理,得到局部凸空间中集值映射的极小不动点定理,应用此定理,证明了半线性不适定的算子方程的最小范数极值解的存在性.此结果可以应用到不适定常微方程的两点边值问题,不适定偏微方程的边值问题.     

On the complexity and approximability of some Euclidean optimal summing problems

The complexity status of several well-known discrete optimization problems with the direction of optimization switching from maximum to minimum is analyzed. The task is to find a subset of a finite set of Euclidean points (vectors). In these problems, the objective functions depend either only on the norm of the sum of the elements from the subset or on this norm and the cardinality of the subset. It is proved that, if the dimension of the space is a part of the input, then all these problems are strongly NP-hard. Additionally, it is shown that, if the space dimension is fixed, then all the problems are NP-hard even for dimension 2 (on a plane) and there are no approximation algorithms with a guaranteed accuracy bound for them unless P = NP. It is shown that, if the coordinates of the input points are integer, then all the problems can be solved in pseudopolynomial time in the case of a fixed space dimension.     

Vector-valued Lg-splines II. Smoothing splines

In the first paper of this series, Lg-spline theory was extended to the vector-valued interpolating case. Here this work is complemented by giving the extension for smoothing splines. The problem is formulated as a constrained minimum norm problem in a reproducing kernel Hilbert space, and solved recursively using a congruent stochastic estimation model.     

A Boundary Value Problem for an Elliptic Equation with Asymmetric Coefficients in a Nonschlicht Domain

We propose some minimum principle for the quadratic energy functional of an elliptic boundary value problem describing a transport process with asymmetric tensor coefficients in a nonschlicht domain. We prove the existence and uniqueness of a weak solution in the energy space. The energy norm equals the entropy production rate.