不确定规划逼近问题最优解的几乎处处上半收敛性

对不确定规划经验逼近问题的最优解的几乎处处上半收敛性进行了研究。首先将带有约束的不确定规划问题转化成与其等价的无约束的不确定优化问题,然后将经验测度替代不确定测度得到不确定规划的经验逼近模型,并得出逼近问题的目标函数序列的几乎处处上图收敛性,最后利用上图收敛性理论,给出了不确定规划经验逼近最优解集的几乎处处上半收敛性。     

随机规划ε-逼近最优解集的Hausdorff收敛性

本文研究了随机规划ε-逼近最优解集的Haudorff收敛性条件,证明了随机规划逼近最优值的收敛性,并利用此结果给出了随机规划ε-逼近最优解集Haudorff收敛的一个充分条件.     

随机环境中受控分枝过程的极限定理

讨论了随机环境中受控分枝过程{Z_n:n∈N}的极限问题.给出了过程在{S_n:n∈N}下的规范化过程{W_n:n∈N}几乎处处收敛、L~1收敛和L~2收敛的充分条件,以及过程{W:n∈N}的极限非退化于0的充分条件和必要条件,得到了过程在{I_n:n∈N}下的规范化过程{W_n:n∈N}几乎处处收敛和L~1收敛的充分条件.     

二层随机规划逼近问题最优解集的上半收敛性

在下层初始随机规划问题可行解集上引入了正则的概念,并在下层初始随机规划最优解唯一的条件下,利用上图收敛理论,给出了下层随机规划逼近问题的任意一个最优解向量函数都连续收敛到下层初始随机规划问题的唯一最优解向量函数.然后将下层随机规划的最优解向量函数反馈到上层随机规划的目标函数和约束条件中,得到了上层随机规划逼近问题的最优解集关于最小信息概率度量收敛的上半收敛性.     

极大极小随机规划逼近问题最优解集和最优值的稳定性

研究了特殊的二层极大极小随机规划逼近收敛问题. 首先将下层初始随机规划最优解集拓展到非单点集情形, 且可行集正则的条件下, 讨论了下层随机规划逼近问题最优解集关于上层决策变量参数的上半收敛性和最优值函数的连续性. 然后把下层随机规划的epsilon-最优解向量函数反馈到上层随机规划的目标函数中, 得到了上层随机规划逼近问题的最优解集关于最小信息概率度量收敛的上半收敛性和最优值的连续性.     

一类随机规划逼近最优值和最优解集的收敛性

本文提出强上图收敛的概念,讨论了逼近随机规划的目标函数序列的强上图收敛性,研究了逼近随机规划最优值和最优解集的收敛性条件,得到了一类随机规划逼近最优值和最优解集的收敛性.     

一类极大极小随机规划逼近问题最优解集的上半收敛性

本文首先将极大极小随机规划等价的转化为一个二层随机规划,在下层初始随机规划最优解集为多点集的情形下,给出下层随机规划逼近问题最优解集集值映射关于上层决策变量参数的上半收敛性和最优值函数的连续性.然后将上层随机规划等价转化为以上层和下层决策变量作为整体决策变量,以下层规划最优解集的图作为约束条件的单层规划,并在下层初始随机规划最优解集的图为正则的条件下,得到上层随机规划逼近问题最优解集关于最小信息概率度量收敛的上半收敛性.     

二层随机规划逼近最优解集的上半收敛性

下层随机规划以上层决策变量作为参数,而上层随机规划是以下层随机规划的唯一最优解作为响应的一类二层随机规划问题,首先在下层随机规划的原问题有唯一最优解的假设下,讨论了下层随机规划的任意一个逼近最优解序列都收敛于原问题的唯一最优解,然后将下层随机规划的唯一最优解反馈到上层,得到了上层随机规划逼近最优解集序列的上半收敛性.     

关于weierstrass逼近定理的几点注记

Weierstrass逼近定理是函数逼近论中的重要定理之一,定理阐述了闭区间上的连续函数可以用一多项式去逼近.将该定理进行推广:即使一个函数是几乎处处连续的,也不一定具有与连续函数相类似的逼近性质,但是一个处处不连续的函数却有可能具有这样的性质.证明了定义在闭区间上且与连续函数几乎处处相等的函数具有类似的逼近性质,并给出了weierstrass逼近定理的一个推广应用.     

一类随机泛函微分方程带随机步长的EM逼近的渐近稳定

研究了一类带有限延迟的随机泛函微分方程的Euler-Maruyama(EM)逼近，给出了该方程的带随机步长的EM算法，得到了随机步长的两个特点:首先，有限个步长求和是停时；其次，可列无限多个步长求和是发散的.最终，由离散形式的非负半鞅收敛定理，得到了在系数满足局部Lipschitz条件和单调条件下，带随机步长的EM数值解几乎处处收敛到0.该文拓展了2017年毛学荣关于无延迟的随机微分方程带随机步长EM数值解的结果.     

Stability in the almost everywhere sense: A linear transfer operator approach

The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.     

Lyapunov density for coupled systems

We prove a necessary and sufficient condition for the existence of Lyapunov density for a system of coupled autonomous ordinary differential equations. In particular, we characterize the kinds of couplings that preserve almost everywhere uniform stability of the origin provided the isolated systems have an almost everywhere uniformly stable equilibrium point at the origin.     

The surprising almost everywhere convergence of Fourier-Neumann series

For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in L^p requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L² is the celebrated Carleson theorem, proved in 1966 (and extended to L^p by Hunt in 1967).In this paper, we take the system

     

随机环境中迁入分枝过程的极限性质

在方差和均值有限的条件下,得到了随机环境中迁入分枝过程对应的规范化过程的几乎处处收敛性和L^2收敛性．这对于刻画过程本身的发散速度,具有重要的意义．     

Convergence rate of Cesàro means of Fourier-Laplace series

The convergence rate of Fourier-Laplace series in logarithmic subclasses of L²(Σd) defined in terms of moduli of continuity is of interest. Lin and Wang [C. Lin, K. Wang, Convergence rate of Fourier-Laplace series of L²-functions, J. Approx. Theory 128 (2004) 103-114] recently presented a characterization of those subclasses and provided the almost everywhere convergence rates of Fourier-Laplace series in those subclasses. In this note, the almost everywhere convergence rates of the Cesàro means for Fourier-Laplace series of the logarithmic subclasses are obtained. The strong approximation order of the Cesàro means and the partial summation operators are also presented.     

Quantitative stability of full random two-stage stochastic programs with recourse

In this paper, we consider quantitative stability analysis for two-stage stochastic linear programs when recourse costs, the technology matrix, the recourse matrix and the right-hand side vector are all random. For this purpose, we first investigate continuity properties of parametric linear programs. After deriving an explicit expression for the upper bound of its feasible solutions, we establish locally Lipschitz continuity of the feasible solution sets of parametric linear programs. These results are then applied to prove continuity of the generalized objective function derived from the full random second-stage recourse problem, from which we derive new forms of quantitative stability results of the optimal value function and the optimal solution set with respect to the Fortet–Mourier probability metric. The obtained results are finally applied to establish asymptotic behavior of an empirical approximation algorithm for full random two-stage stochastic programs.     

Recourse-based stochastic nonlinear programming: properties and Benders-SQP algorithms

In this paper, we study recourse-based stochastic nonlinear programs and make two sets of contributions. The first set assumes general probability spaces and provides a deeper understanding of feasibility and recourse in stochastic nonlinear programs. A sufficient condition, for equality between the sets of feasible first-stage decisions arising from two different interpretations of almost sure feasibility, is provided. This condition is an extension to nonlinear settings of the “W-condition,” first suggested by Walkup and Wets (SIAM J. Appl. Math. 15:1299–1314, 1967). Notions of complete and relatively-complete recourse for nonlinear stochastic programs are defined and simple sufficient conditions for these to hold are given. Implications of these results on the L-shaped method are discussed. Our second set of contributions lies in the construction of a scalable, superlinearly convergent method for solving this class of problems, under the setting of a finite sample-space. We present a novel hybrid algorithm that combines sequential quadratic programming (SQP) and Benders decomposition. In this framework, the resulting quadratic programming approximations while arbitrarily large, are observed to be two-period stochastic quadratic programs (QPs) and are solved through two variants of Benders decomposition. The first is based on an inexact-cut L-shaped method for stochastic quadratic programming while the second is a quadratic extension to a trust-region method suggested by Linderoth and Wright (Comput. Optim. Appl. 24:207–250, 2003). Obtaining Lagrange multiplier estimates in this framework poses a unique challenge and are shown to be cheaply obtainable through the solution of a single low-dimensional QP. Globalization of the method is achieved through a parallelizable linesearch procedure. Finally, the efficiency and scalability of the algorithm are demonstrated on a set of stochastic nonlinear programming test problems.     

Periodic decomposition of measurable integer valued functions

We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough. We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so-called decomposition property. We characterize those periods a₁,…,ak for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a₁,…,ak)-periodic decomposition if and only if Δa₁?Δakf=0, where Δaf(x)=f(x+a)−f(x).     

Multivariate robust second-order stochastic dominance and resulting risk-averse optimization

ABSTRACT

By utilizing a min-biaffine scalarization function, we define the multivariate robust second-order stochastic dominance relationship to flexibly compare two random vectors. We discuss the basic properties of the multivariate robust second-order stochastic dominance and relate it to the nonpositiveness of a functional which is continuous and subdifferentiable everywhere. We study a stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and develop the necessary and sufficient conditions of optimality in the convex case. After specifying an ambiguity set based on moments information, we approximate the ambiguity set by a series of sets consisting of discrete distributions. Furthermore, we design a convex approximation to the proposed stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and establish its qualitative stability under Kantorovich metric and pseudo metric, respectively. All these results lay a theoretical foundation for the modelling and solution of complex stochastic decision-making problems with multivariate robust second-order stochastic dominance constraints.