变分不等式的极大熵函数方法

利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题,给出了不等式组及变分不等式问题近似解的可微优化方法,得到了不等式组和变分不等式问题的解集合的示性函数.     

不等式组与变分不等式的极大熵函数方法

利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题，给出了不等式组及变分不等式问题近似解的可微优化方法，得到了不等式组和变分不等式问题的解集合的示性函数．     

中点不等式和梯形不等式的量子模拟

建立了涉及二阶q-导数的积分恒等式,使用这个恒等式获得二阶可微函数中点不等式和梯形不等式的量子模拟.     

高阶可微函数Ostrowski型不等式

研究了一类n阶可微函数,利用其n阶导数上、下界以及Cruis不等式,给出了n阶可微函数Ostrowski型不等式,从而推广二阶可微函数Ostrowski型不等式.     

非线性不等式系统带折线步的信赖域方法及其收敛性

针对一类非线性不等式系统求解的问题,利用一系列目标函数二次可微的带参数优化问题来逐次逼近非线性不等式系统的解,从而提出了针对参数最优化问题带折线步的信赖域算法．在较弱的条件下,算法的全局收敛性得到了保证．数值试验显示算法有效．     

不等式的解法

考点评析不等式的解法仍是高考命题的热点之一 ,不等式的有关内容仍将在函数、数列、几何、实际应用等有关的综合题中考查 .1.1　知识点剖析在熟练掌握一元一次不等式 (组 )、一元二次不等式的解法基础上初步掌握其他一些简单的不等式的解法 ,如高次不等式、分式不等式、无理不等式、含绝对值的不等式、指数不等式和对数不等式的求解 ,一般是将它们进行同解变换 (即等价变换 )化为一元一次不等式 (组 )或一元二次不等式 (组 )后而得其解 .要注意对含字母系数的不等式须经讨论求解的问题 .1.2　思想方法化 (无理 )为有理 ,化 (分式 )为整式…     

关于非线性不等式组Levenberg-Marquardt算法的收敛性(英文)

本文研究了一类非线性不等式组的求解问题.利用一列目标函数两次可微的参数优化问题来逼近非线性不等式组的解,光滑Levenberg-Marquardt方法来求解参数优化问题,在一些较弱的条件下证明了文中算法的全局收敛性,数值实例显示文中算法效果较好.     

化不等式约束问题为无约束的广义可微限域精确罚函数方法

本文给出了广义可微精确罚函数的概念及一类所谓广义限域可微精确罚函数.本文预先选定罚因子,将不等式约束问题化为单一的无约束问题,并给出了具全局收敛性的算法.本文的罚函数构造简单,假设条件少而且算法的构造与收敛性结果是独特的.     

一类非线性积分不等式组中未知函数的估计

研究了一类二维非线性积分不等式组,该不等式组积分号外有非常数因子,不能用向量形式的Gronwall-Bellman型积分不等式进行估计.先利用Bernoulli不等式把非线性问题转化成线性问题,利用变量替换技巧和放大技巧研究只含有一个未知函数的积分不等式,接着利用两个引理和变量替换技巧和放大技巧给出不等式组中两个未知函数的估计.结果可用于研究积分、微分动力系统解的性质.     

追求本质简单自然的数学教学——“一元二次不等式的解法”教学设计与思考

一、课题分析 “一元二次不等式的解法”具有以下三个特点： 1．一元二次不等式的解法是一元一次不等式的解法的延续和深化,它对集合知识起到重要的巩固和运用的作用,也与后继的函数、三角函数、线性规划、直线与圆锥曲线以及导数等内容紧密相关．许多问题的解决都要建立在一元二次不等式正确求解的基础上．可见,一元二次不等式的解法在高中数学中具有极强的基础性和工具性;     

三阶可微函数的若干不等式

通过建立与Bullen不等式有关的积分恒等式,利用Halder不等式、Grüss不等式、Chebychev不等式,给出三阶可微函数的一些不等式.     

On refinements of some integral inequalities using improved power-mean integral inequalities

In this study, using power-mean inequality and improved power-mean integral inequality better approach than power-mean inequality and an identity for differentiable functions, we get inequalities for functions whose derivatives in absolute value at certain power are convex. Numerically, it is shown that improved power-mean integral inequality gives better approach than power-mean inequality. Some applications to special means of real numbers and some error estimates for the midpoint formula are also given.     

On a theorem due to Crouzeix and Ferland

In this article we introduce the notions of Kuhn-Tucker and Fritz John pseudoconvex nonlinear programming problems with inequality constraints. We derive several properties of these problems. We prove that the problem with quasiconvex data is (second-order) Kuhn-Tucker pseudoconvex if and only if every (second-order) Kuhn-Tucker stationary point is a global minimizer. We obtain respective results for Fritz John pseudoconvex problems. For the first-order case we consider Fréchet differentiable functions and locally Lipschitz ones, for the second-order case Fréchet and twice directionally differentiable functions.     

Second-Order Optimality Conditions in Generalized Semi-Infinite Programming

This paper deals with generalized semi-infinite optimization problems where the (infinite) index set of inequality constraints depends on the state variables and all involved functions are twice continuously differentiable. Necessary and sufficient second-order optimality conditions for such problems are derived under assumptions which imply that the corresponding optimal value function is second-order (parabolically) directionally differentiable and second-order epiregular at the considered point. These sufficient conditions are, in particular, equivalent to the second-order growth condition.     

A dual differentiable exact penalty function

A new penalty function is associated with an inequality constrained nonlinear programming problem via its dual. This penalty function is globally differentiable if the functions defining the original problem are twice globally differentiable. In addition, the penalty parameter remains finite. This approach reduces the original problem to a simple problem of maximizing a globally differentiable function on the product space of a Euclidean space and the nonnegative orthant of another Euclidean space. Many efficient algorithms exist for solving this problem. For the case of quadratic programming, the penalty function problem can be solved effectively by successive overrelaxation (SOR) methods which can handle huge problems while preserving sparsity features. Sponsored by the United States Army under Contract No. DAAG 29-80-C-0041. This material is based upon work supported by the National Science Foundation under Grants No. MCS-790166 and ENG-7903881.     

Second-order global optimality conditions for convex composite optimization

In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Convexity results and sharp error estimates in approximate multivariate integration

An interesting property of the midpoint rule and the trapezoidal rule, which is expressed by the so-called Hermite-Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite-Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. In particular, for simplices of arbitrary dimension, we present two families of integration formulae which both contain a multivariate analogue of the midpoint rule and the trapezoidal rule as boundary cases. The first family also includes a multivariate analogue of a Maclaurin formula and of the two-point Gaussian quadrature formula; the second family includes a multivariate analogue of a formula by P.C. Hammer and of Simpson's rule. In both families, we trace out those formulae which satisfy a Hermite-Hadamard inequality. As an immediate consequence of the latter, we obtain sharp error estimates for twice continuously differentiable functions.

     

Fejér不等式加强形式的推广

通过建立积分恒等式,将一个关于4阶可微函数的Fejér不等式的加强形式推广到关于2n-1(n≥2)阶可微函数的Fejér型不等式.对此新推广的不等式还给出了若干误差估计.     

Strong Geodesic Convex Functions of Order m

Abstract

Strong geodesic convex function and strong monotone vector field of order m on Riemannian manifolds are established. A characterization of strong geodesic convex function of order m for the continuously differentiable functions is discussed. The relation between the solution of a new variational inequality problem and the strict minimizers of order m for a multiobjective programing problem is also established.