概率约束规划的稳定性分析

本文对概率约束规划问题的稳定性进行了探讨,得出了当随机向量序列{ξ^(k)(ω）}分布收敛于ξ（ω）时,相应于ξ^(k)(ω)的概率约束规划问题的最优值收敛于原问题的最优值,这个结果为设计逼近算法和改进逼近解提供了一个理论基础。     

二阶泛函微分方程的渐近性和振动性

本文讨论了二阶泛函微分方程的解的渐近性和振动性。文中指出,当sum from to +∞ (g^-1)(1/(r(t))dt<+∞时,(1)式的非振动解的渐近性态有且仅有如下的四种类型:A_c^k,A_c^∞,A_o^k,A_o^∞。当sum from to +∞ (g^-1)(1/(r(t))dt=+∞时,(1)式的非振动解的渐近性态有且仅有如下三种类型:A_c^o,A_∞^c,A_∞^c。在f为超线性或次线性的前提下,本文分别给出了存在A_c^k,A_c^∞,A_o^k,A_c^o,A_∞^c等型非振动解的充要条件。在f为强超线性或强次线性的前提下,本文给出了方程(1)为振动的充要条件。     

关于R(Ax)闭的连续算子族Ax的Moore-Penrose广义逆Ax+连续的充要条件

设X是拓扑空间,A_x,闭值域R(A_x)为X→B(H)的连续映射,我们知道:即使在dim(H)有限的情况下,M.-P.逆A_x⁺:也不一定连续,A_x⁺连续的充要条件,对现代分析和应用数学的一些研究是很重要的,本文给出了,在一般拓扑空间X、局部紧拓扑空间X、A_x为Fredholm算子族等情况下,A_x⁺为连续的充分必要条件。     

关于概率约束条件线性规划的一个讨论结果

本文给出概率约束规划问题Ｐ：ｍｉｎ｛ｃｘ｜Ｐ（Ａ１ｘ≥ξ）≥ｐ，Ａ２ｘ≥ｂ｝最优解的充分必要条件，其中ξ是连续随机变量，ｐ是置信水平（０＜ｐ＜１）     

满足?A0∈PA(VA0、V(A-A0)∈B→GA0=φ)的可数无穷非原子布尔代数的相对结构以及对可数无穷非原子布尔代数结构的探讨

本文讨论了满足?A₀∈PA(VA₀、V(A-A₀)∈B→G_A0=φ)的可数无穷非原子布尔代数与原子、无原子布尔代数在结构上的关系,给出了任意两个这种类型的布尔代数同构的充要条件,并且对可数无穷非原子布尔代数的结构进行了探讨。     

Brouwer不动点定理与顶点在曲面上的四边形

设D是Euclid平面R² 中的一个半径为r的圆盘 ,F是D上Lipschitz连续的实值函数 ,A₁A₂ A₃ A₄是边长不超过r的等腰梯形 ,∠A₁A₂ A₃ =α≤π/ 2 .借助于Brouwer不动点定理证明了 :若F有一个Lipschitz常数λ≤min{1 ,tgα},则在曲面M ={(x ,y ,F(x ,y) )∈R³ ∶(x ,y)∈D}上存在共平面的四个点 ,它们张成一个与A₁A₂ A₃ A₄全等的四边形 .此外 ,还作了一些进一步的讨论 .     

倾斜代数表示型的一个判别

给定遗传代数A和倾斜模_AT.记 B=End_AT为相应的倾斜代数。本文给出一种约简程序,得到两个遗传代数_sA和A_t;证明了,B是有限表示型当且仅当_sA=0和A_t=0,并且B分别是tame型和 wild型当且仅当代数直积_sA?A_t分别是tame型和wild型。     

概率密度的均匀核估计与近邻估计的均方误差

In this paper we consider the Mean Square Error (MSE) of two uaual estimates of density function f(x) at a point x: The uniform kernel estimate f_n(x) and the NN estimate fn(x). we- show that when f is differentiable for sufficiently high order at x. these MSE can be expanded in a form E(f_n(x)-f(x))²=A₁(x)n^-4/5 +A₂(x)n^-1+A₃(x)n^-6/5+…;E(f_n(x)-f(x))²=B₁(x)n^-4/5 +B₂(x)n^-1+B₃(x)n^-6/5+… And if we suitably choose the parameters in f_n and f_n to make A₁(x) and B₁(x)to assume its minimunm value, then we also, have A₂(x) =B₂(x) but A₃(X) differs form B₃(X). This result shows that while the two estimates are not identical with respect to MSE. each one can be superior to the other in various special cases.     

解在QK 型及Dirichlet 型空间中的线性微分方程

对于单位圆盘上系数函数是解析函数的复微分方程
f⁽ⁿ⁾+A_n-1(z)f^(n-1)+…+A₁(z)f''+A₀(z)f=0,
给出了方程的系数函数和解函数之间的关系, 即当系数函数A_j 满足给定的条件时, 方程的所有解属于Q_K型空间和Dirichlet 型空间.     

赋权矩阵及其性质

本文给出了以S为行权和向量的权矩阵类T_＋(S)中每个权矩阵都可逆的一个充要条件．     

Lagrange Multiplier Conditions Characterizing the Optimal Solution Sets of Cone-Constrained Convex Programs

Various characterizations of optimal solution sets of cone-constrained convex optimization problems are given. The results are expressed in terms of subgradients and Lagrange multipliers. We establish first that the Lagrangian function of a convex program is constant on the optimal solution set. This elementary property is then used to derive various simple Lagrange multiplier-based characterizations of the solution set. For a finite-dimensional convex program with inequality constraints, the characterizations illustrate that the active constraints with positive Lagrange multipliers at an optimal solution remain active at all optimal solutions of the program. The results are applied to derive corresponding Lagrange multiplier characterizations of the solution sets of semidefinite programs and fractional programs. Specific examples are given to illustrate the nature of the results.     

On characterizing the solution sets of pseudolinear programs

This paper provides several new and simple characterizations of the solution sets of pseudolinear programs. By means of the basic properties of pseudolinearity, the solution set of a pseudolinear program is characterized, for instance, by the equality that , for each feasible pointx, where is in the solution set. As a consequence, we give characterizations of both the solution set and the boundedness of the solution set of a linear fractional program.     

An extension of the simplex algorithm for semi-infinite linear programming

We present a primal method for the solution of the semi-infinite linear programming problem with constraint index setS. We begin with a detailed treatment of the case whenS is a closed line interval in . A characterization of the extreme points of the feasible set is given, together with a purification algorithm which constructs an extreme point from any initial feasible solution. The set of points inS where the constraints are active is crucial to the development we give. In the non-degenerate case, the descent step for the new algorithm takes one of two forms: either an active point is dropped, or an active point is perturbed to the left or right. We also discuss the form of the algorithm when the extreme point solution is degenerate, and in the general case when the constraint index set lies in ^p . The method has associated with it some numerical difficulties which are at present unresolved. Hence it is primarily of interest in the theoretical context of infinite-dimensional extensions of the simplex algorithm.     

On the continuity of the minimum in parametric quadratic programs

This paper establishes results on the lower semicontinuity and continuity of the optimal objective value of a parametric quadratic program with continuous dependence of the coefficients on parameters. The results established herein generalize directly well-known results on parametric linear programs.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A8189.     

A Reformulation-Linearization Technique (RLT) for semi-infinite and convex programs under mixed 0-1 and general discrete restrictions

The Reformulation-Linearization Technique (RLT) provides a hierarchy of relaxations spanning the spectrum from the continuous relaxation to the convex hull representation for linear 0-1 mixed-integer and general mixed-discrete programs. We show in this paper that this result holds identically for semi-infinite programs of this type. As a consequence, we extend the RLT methodology to describe a construct for generating a hierarchy of relaxations leading to the convex hull representation for bounded 0-1 mixed-integer and general mixed-discrete convex programs, using an equivalent semi-infinite linearized representation for such problems as an intermediate stepping stone in the analysis. For particular use in practice, we provide specialized forms of the resulting first-level RLT formulation for such mixed 0-1 and discrete convex programs, and illustrate these forms through two examples.     

Generalized proximal point algorithm for convex optimization

Ha (Ref. 1) recently introduced a generalized proximal point algorithm for solving a generalized equation. In this note, we present a generalized proximal point algorithm for convex optimization problems based on Ha's work. The idea behind this algorithm is that, instead of adding a quadratic term to all the variables, we add a quadratic term to a subset of the variables. We extend the criteria for approximate solutions given by Rockafellar (Ref. 2) and Auslender (Ref. 3) and present convergence results. Finally, we show how this algorithm can be applied to solve block-angular linear and quadratic programming problems.     

A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique

This paper is concerned with the development of an algorithm to solve continuous polynomial programming problems for which the objective function and the constraints are specified polynomials. A linear programming relaxation is derived for the problem based on a Reformulation Linearization Technique (RLT), which generates nonlinear (polynomial) implied constraints to be included in the original problem, and subsequently linearizes the resulting problem by defining new variables, one for each distinct polynomial term. This construct is then used to obtain lower bounds in the context of a proposed branch and bound scheme, which is proven to converge to a global optimal solution. A numerical example is presented to illustrate the proposed algorithm.     

A generalization of the construction of test problems for nonconvex optimization

In this paper we adopt and generalize the basic idea of the method presented in [3] and [4] to construct test problems that involve arbitrary, not necessarily quadratic, concave functions, for both Concave Minimization and Reverse Convex Programs     

Unconstrained convex programming approach to linear programming

Recently, Fang proposed approximating a linear program in the Karmarkar standard form by adding an entropic barrier function to the objective function and derived an unconstrained dual concave program. We present in this note a necessary and sufficient condition for the existence of a dual optimal solution to the perturbed problem. In addition, a sharp upper bound of error estimation in this approximation scheme is provided.     

Equivalence between a generalized fenchel duality theorem and a saddle-point theorem for fractional programs

The convex case of a fractional program is considered. By the aid of a duality theory for mathematical programming involving a maximum and minimum operator, and by defining new operators, we obtain three equivalent duality theorems for three pairs of primal-dual programs. The second and the third one are the saddle-point theorem and a generalized Fenchel duality theorem, respectively.