Turnpike Properties for Stochastic Linear-Quadratic Optimal Control Problems*

This paper analyzes the limiting behavior of stochastic linear-quadratic optimal control problems in finite time-horizon [0, T ] as T → ∞. The so-called turnpike properties are established for such problems, under stabilizability condition which is weaker than the controllability, normally imposed in the similar problem for ordinary differential systems.In dealing with the turnpike problem, a crucial issue is to determine the corresponding static optimization problem. Intuitively mimicking the deterministic situations, it seems to be natural to include both the drift and the diffusion expressions of the state equation to be zero as constraints in the static optimization problem. However, this would lead us to a wrong direction. It is found that the correct static problem should contain the diffusion as a part of the objective function, which reveals a deep feature of the stochastic turnpike problem.     

Perturbation analysis and condition numbers of scaled total least squares problems

The standard approaches to solving an overdetermined linear system Ax ≈ b find minimal corrections to the vector b and/or the matrix A such that the corrected system is consistent, such as the least squares (LS), the data least squares (DLS) and the total least squares (TLS). The scaled total least squares (STLS) method unifies the LS, DLS and TLS methods. The classical normwise condition numbers for the LS problem have been widely studied. However, there are no such similar results for the TLS and the STLS problems. In this paper, we first present a perturbation analysis of the STLS problem, which is a generalization of the TLS problem, and give a normwise condition number for the STLS problem. Different from normwise condition numbers, which measure the sizes of both input perturbations and output errors using some norms, componentwise condition numbers take into account the relation of each data component, and possible data sparsity. Then in this paper we give explicit expressions for the estimates of the mixed and componentwise condition numbers for the STLS problem. Since the TLS problem is a special case of the STLS problem, the condition numbers for the TLS problem follow immediately from our STLS results. All the discussions in this paper are under the Golub-Van Loan condition for the existence and uniqueness of the STLS solution. Yimin Wei is supported by the National Natural Science Foundation of China under grant 10871051, Shanghai Science & Technology Committee under grant 08DZ2271900 and Shanghai Education Committee under grant 08SG01. Sanzheng Qiao is partially supported by Shanghai Key Laboratory of Contemporary Applied Mathematics of Fudan University during his visiting.     

Riesz界与条件数

在多分辨分析与小波分析中,人们经常要估计Riesz基的上下界.在有限维空间中,这等价于计算Riesz基所对应Gramian矩阵的条件数.本文给出Riesz基与条件数的关系并且讨论了Riesz基加入元素后对Riesz界产生的影响.     

On Condition Number Theorems in Mathematical Programming

A condition number of nonconvex mathematical programming problems is defined as a measure of the sensitivity of their global optimal solutions under canonical perturbations. A (pseudo-)distance among problems is defined via the corresponding augmented Kojima functions. A characterization of well-conditioning is obtained. In the nonconvex case, we prove that the distance from ill-conditioning is bounded from above by a multiple of the reciprocal of the condition number. Moreover, a lower bound of the distance from a special class of ill-conditioned problems is obtained in terms of the condition number. The proof is based on a new theorem about the permanence of the Lipschitz character of set-valued inverse mappings. A uniform version of the condition number theorem is proved for classes of convex problems defined through bounds of some constants available from problem’s data.     

追加数据对条件指数的影响<英>

克服线性回归模型中设计矩阵的复共线性,主要有三种方法:追加数据,自变量选择和非最小二乘法,本文研究追加数据在减少条件数中的作用,我们的研究表明,在可能情况下适当地选择追加数据,设计矩阵的条件指数可以被减少.我们用在回归最优设计中广泛被研究过的Gaylor-Merrill数据说明了本文理论结果的实用意义.     

Criteria for Unconstrained Global Optimization

We develop criteria for the existence and uniqueness of the global minima of a continuous bounded function on a noncompact set. Special attention is given to the problem of parameter estimation via minimization of the sum of squares in nonlinear regression and maximum likelihood. Definitions of local convexity and unimodality are given using the level set. A fundamental theorem of nonconvex optimization is formulated: If a function approaches the minimal limiting value at the boundary of the optimization domain from below and its Hessian matrix is positive definite at the point where the gradient vanishes, then the function has a unique minimum. It is shown that the local convexity level of the sum of squares is equal to the minimal squared radius of the regression curvature. A new multimodal function is introduced, the decomposition function, which can be represented as the composition of a convex function and a nonlinear function from the argument space to a space of larger dimension. Several general global criteria based on majorization and minorization functions are formulated.     

On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems

Classical condition numbers are normwise: they measure the size of both input perturbations and output errors using some norms. To take into account the relative of each data component, and, in particular, a possible data sparseness, componentwise condition numbers have been increasingly considered. These are mostly of two kinds: mixed and componentwise. In this paper, we give explicit expressions, computable from the data, for the mixed and componentwise condition numbers for the computation of the Moore-Penrose inverse as well as for the computation of solutions and residues of linear least squares problems. In both cases the data matrices have full column (row) rank.

     

Mean-Value Theorems in Arithmetic Semigroups

This paper reports on recent progress in the theory of multiplicative arithmetic semigroups, which has been initiated by John Knopfmacher's work on abstract analytic number theory. In particular, it deals with abstract versions of the mean-value theorems of Delange, of Wirsing, and of Halász for multiplicative functions on arithmetic semigroups G with Axiom A . The Turán Kubilius inequality is transferred to G , and methods developed by Rényi, Daboussi and Indlekofer, Lucht and Reifenrath are utilized. As byproduct a new proof of the abstract prime number theorem is obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Selection of One Feedback Nash Equilibrium in Discounted Linear-Quadratic Games

We study a selection method for a Nash feedback equilibrium of a one-dimensional linear-quadratic nonzero-sum game over an infinite horizon. By introducing a change in the time variable, one obtains an associated game over a finite horizon T > 0 and with free terminal state. This associated game admits a unique solution which converges to a particular Nash feedback equilibrium of the original problem as the horizon T goes to infinity.     

Theorems of the Alternative and Optimization with Set-Valued Maps

In this paper, the concept of generalized cone subconvexlike set-valued mapsis presented and a theorem of alternative for the system of generalizedinequality–equality set-valued maps is established. By applying thetheorem of the alternative and other results, necessary and sufficientoptimality conditions for vector optimization problems with generalizedcone subconvexlike set-valued maps are obtained.     

YoshiZawa型周期解定理和Massera型周期解定理研究进展简介

微分方程解的有界性和周期解的存在性是檄分方程理论研究中的两个重要课题，二者之间有着紧密的联系．在解的有界性与周期解的存在性的研究中，Yoshizawa周期解定理和Massera周期解定理是非常重要的结果，具有重要的理论意义和应用价值．本文以Yoshizawa型周期解定理和Massera型周期解定理的研究为主，简要介绍泛函微分方程周期解理论研究方面的一些新进展。     

On the condition number of linear least squares problems in a weighted Frobenius norm

LetA be anm × n, m n full rank real matrix andb a real vector of sizem. We give in this paper an explicit formula for the condition number of the linear least squares problem (LLSP) defined by min Ax–b₂,x ⁿ . Let and be two positive real numbers, we choose the weighted Frobenius norm [A, b] _F on the data and the usual Euclidean norm on the solution. A straightforward generalization of the backward error of [9] to this norm is also provided. This allows us to carry out a first order estimate of the forward error for the LLSP with this norm. This enables us to perform a complete backward error analysis in the chosen norms.Finally, some numerical results are presented in the last section on matrices from the collection of [5]. Three algorithms have been tested: the QR factorization, the Normal Equations (NE), the Semi-Normal Equations (SNE).     

匹配数为2的图的最小特征值

在所有给定阶数且匹配数为2的连通图中,我们刻画了最小特征值达到极小的图,给出了这类图最小特征值的下界.     

Antipodal Theorems for Sets in Rn Bounded by a Finite Number of Spheres

This is a continuation of paper in Adv. Appl. Math. 22 (1999), 219–226, on an antipodal theorem for sets Dⁿ in Rⁿ bounded by a finite number of spheres. Here this theorem is first applied to set-valued mappings from Dⁿ to the boundary of an (n + 1)-cube or a d- dimensional octahedron. Next, the antipodal theorem is reformulated in terms of real continuous functions on Dⁿ, together with applications to the classical theorems of Borsuk–Ulam and Lusternik–Schnirelmann–Borsuk.     

实数系基本定理等价性的完全互证

综合给出了实数系六个基本定理的等价性的完全互证方法,并归纳了各种证明方法的规律,旨在把抽象的证明转化为容易掌握的基本方法.     

CONDITION NUMBER FOR WEIGHTED LINEAR LEAST SQUARES PROBLEM

In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem： minx ｜｜Ax- b｜｜2, where A is an m-by-n （m ≥ n） rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm ｜｜ [AT,βb] ｜｜F of the data A and b, where T is a positive diagonal matrix and β is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.     

On Efficient Solutions in Vector Optimization

A new characterization is obtained for the existence of an efficient solution of a vector optimization problem in terms of associated scalar optimization problems. The consequences for linear vector optimization problems are derived as a special case, Applications to convex vector optimization problems are also discussed.     

G-凸空间中的KKM定理，匹配定理和截口定理

在G-凸空间中证明了一些新的KKM型定理．作为应用,在G-凸空间中得到了一些新的匹配定理和截口定理,所得结果改进和推广了[2,3,7]中的相关结果．     

一个图的匹配数条件

设G是一个简单图,在图G中任意一个最大匹配的基数叫做G的匹配数,记作v(G),在这篇文章中我们获得了下面的结果,(1)设G是连通的和不完全的,则对于x,y∈v(G)和xyE(G),v(G-{x,y}=v(G)-1的充分必要条件是(a)G[A(G)]是完全的和A(G)的每一个点和C(G)的每一个点相邻,(b)c(D(G))=|A(G)| 1,和(c)y∈D(G-x)对于x,y∈C(G)。(2)设G是连通的和不完全的,则v(G-{x,y})=v(G)-2对于x,y∈V(G)和xyE(G)的充分必要条件是GK_(n,n),其中n≥2。