非线性规划问题的灵敏度分析和稳定性分析

1.引言在大多数实际问题中,当求解一个非线性规划问题时,我们不但要求得到它的解,而且还要了解当问题随某些参数变化时它的最优解将发生怎样的变化,这种随参数的变化可能是离散的,也可能是连续的。因此我们要考虑最优解的局部扰动(灵敏度)分析和有限扰动(即稳定性)分析。扰动和逼近已经成为数学的重要分支。在数学规划中,灵敏度分析和稳定性分析也已经被用来得到最优性条件,对偶性结果,计算方法的设计,收敛性和敛速估计等等。     

运量分布与交通配流组合问题灵敏度分析方法

主要研究含单边约束的运量分布与交通配流组合问题的灵敏度分析计算方法.通过将该问题的数学规划模型等价转化为变分不等式模型,进而利用变分不等式问题的灵敏度分析方法,得到该组合问题中各决策变量关于扰动参数的导数公式.最后给出一个简单的数值算例说明该灵敏度分析方法的有效性.     

加权线性支持向量分类机的数据扰动分析

加权线性支持向量分类机是数据挖掘的新方法.它对应于一个优化问题.针对加权线性支持向量分类机优化问题建立了数据扰动分析理论方法.具体地针对加权线性支持向量分类机的原始问题建立了数据扰动分析基本定理,定理可以得到加权线性支持向量分类机问题的解及决策函数对数据参数的偏导数,同时可以定量分析输入数据的误差以及数据各种变化对其解以及决策函数值的定量影响,可以回答加权线性支持向量分类机问题的稳定性问题和灵敏度分析问题.     

线性ε-支持向量回归机的数据扰动分析

支持向量回归机是解决回归问题的一个重要方法.在实际问题中由于测量及计算误差的存在,我们得到的数据往往只是真值的某种近似,带有一定的舍入误差,因此有必要研究支持向量回归机的数据扰动问题.考虑到线性回归问题在实际生活中有广泛的应用价值,把线性ε-支持向量回归机作为研究对象.由于最终关心的是它的原始问题的解,所以我们研究给定的训练集中输入数据发生微小地扰动后,原始问题的解的变化情况.在一定的条件下给出了解对扰动数据偏导数的表达式,建立了线性ε-支持向量回归机的原始问题的灵敏度分析定理.文中还进一步分析了建立该灵敏度分析定理所需要的条件,给出了条件减弱后的结果.文章最后还通过一些简单的数值试验验证了定理的准确性.     

极小扰动约束非线性规划的最优性条件

考虑当目标函数在约束条件下的最优值作扰动时，使各约束作极小扰动的非线性规划问题．文中引进了极小扰动约束规划的极小扰动有效解概念．利用把问题归为一个相应的多目标规划问题，给出了极小扰动约束有效解的最优性条件．     

不存在无风险资产的投资组合灵敏度分析

本文研究了M－V证券投资组合灵敏度分析方法。考虑了不存在无险资产时证券预睡益率和协方差矩阵存在扰动的情形，给出了最优投资组合有效边缘的漂移方程及组合扩展路径。     

接触表面中多场耦合方程的边界层特征解

介绍由约束场和受重力影响的对流扰动耦合而成的衰减平衡向量场动力学方程的渐近求解.为分析实验室内微观与自然界中宏观现象的正则和奇异扰动问题.运用复合尺度方法进行Fourier调和分析、尺度变化,并引进新的参数,将一个复杂的三维约束耦合动力学方程降维投影并转化成复空间里一维的边界层问题.通过渐近摄动分析,给出多场耦合中扰动问题的特征函数边界层解法,在例2中对流场扰动问题分析,得出从指数振荡解过渡到代数解的转点.进一步分析计算非线性特征值问题并做了渐近摄动分析,最后给出多场耦合中扰动问题的特征值边界层解法.最后,特征关系式的各参数表明其在接触表面中对动力衰变的关键影响.     

一种带有生产费用扰动的供应链协调问题

本文研究了一个生产两个竞争零售商的供应链在生产费用发生扰动下的协调问题．生产费用发生扰动后，零售商的投资额与生产商的生产数量都会和原计划不同．研究发现批发价加投资补贴率的合同可以协调扰动后的供应链，通过调整批发价和补贴率，可使供应链总的利润在生产商与两个零售商之间分配，并用数值模拟分析了所得结果的合理性．     

C^n中子空间之间极端主角的一些应用

本文给出了Ｃ＾ｎ中子空间之间最大和最小主角在矩阵逼近，投影算子的扰动分析，群逆以及Ｏｒａｘｉｎ逆的扰动估计，条件数理论，Ｂｏｔｔ－Ｄｕｆｆｉｎ系统扰动分析中一些应用。     

VRPTW的扰动恢复及其TABU SEARCH算法

本文对带时间窗的车辆路线安排扰动恢复问题进行了讨论,分析了各种可能的扰动:增加减少客户,时间窗、客户需求及路线可行性的扰动,构造了扰动模型.利用禁忌搜索算法对问题进行求解,同时通过对模型参数重新设置,得到了多个满足要求的不同的解,这样使解更具有实际可行性和有效性.     

Quasidifferentiability in nonsmooth,nonconvex mechanics

Nonconvex and nonsmooth optimization problems arise in advanced engineering analysis and structural analysis applications. In fact the set of inequality and complementarity relations that describe the structural analysis problem are generated as optimality conditions by the quasidifferential potential energy optimization problem. Thus new kind of variational expressions arise for these problems, which generalize the classical variational equations of smooth mechanics, the variational inequalities of convex, nonsmooth mechanics and give a solid, computationally efficient explication of hemivariational inequalities of nonconvex, nonsmooth mechanics. Moreover quasidifferential calculus and optimization software make this approach applicable for a large number of problems. The connection of quasidifferential optimization and nonsmooth, nonconvex mechanics is discussed in this paper. A number of representative examples from elastostatic analysis applications are treated in details. Numerical examples illustrate the theory.     

Computing Optimal Distances to Pareto Sets of Multi-Objective Optimization Problems in Asymmetric Normed Lattices

Given a finite dimensional asymmetric normed lattice, we provide explicit formulae for the optimization of the associated (non-Hausdorff) asymmetric “distance” among a subset and a point. Our analysis has its roots and finds its applications in the current development of effective algorithms for multi-objective optimization programs. We are interested in providing the fundamental theoretical results for the associated convex analysis, fixing in this way the framework for this new optimization tool. The fact that the associated topology is not Hausdorff forces us to define a new setting and to use a new point of view for this analysis. Existence and uniqueness theorems for this optimization are shown. Our main result is the translation of the original abstract optimal distance problem to a clear optimization scheme. Actually, this justifies the algorithms and shows new aspects of the numerical and computational methods that have been already used in visualization of multi-objective optimization problems.

     

回归分析在洋芋丰产栽培模式研究中的应用

采用三因子五水平二次回归通用旋转组合设计试验,应用回归分析方法建立了洋芋丰产栽培技术模型,并进行了因子效应、交互效应、综合农艺措施寻优等方面的分析,获得高产优化农艺方案。     

Measure concentration in optimization

We discuss some consequences of the measure concentration phenomenon for optimization and computational problems. Topics include average case analysis in optimization, efficient approximate counting, computation of mixed discriminants and permanents, and semidefinite relaxation in quadratic programming.     

Asymptotic analysis in convex composite multiobjective optimization problems

In this paper, we present a unified approach for studying convex composite multiobjective optimization problems via asymptotic analysis. We characterize the nonemptiness and compactness of the weak Pareto optimal solution sets for a convex composite multiobjective optimization problem. Then, we employ the obtained results to propose a class of proximal-type methods for solving the convex composite multiobjective optimization problem, and carry out their convergence analysis under some mild conditions.     

张量分析和多项式优化的若干进展

张量分析 (也称多重数值线性代数) 主要包括张量分解和张量特征值的理论和算法,多项式优化主要包括目标和约束均为多项式的一类优化问题的理论和算法. 主要介绍这两个研究领域中若干新的研究结果. 对张量分析部分,主要介绍非负张量H-特征值谱半径的一些性质及求解方法,还介绍非负张量最大 (小) Z-特征值的优化表示及其解法;对多项式优化部分,主要介绍带单位球约束或离散二分单位取值、目标函数为齐次多项式的优化问题及其推广形式的多项式优化问题和半定松弛解法. 最后对所介绍领域的发展趋势做了预测和展望.     

Performance analysis of distributed solution approaches in simulation-based optimization

Applying computationally expensive simulations in design or process optimization results in long-running solution processes even when using a state-of-the-art distributed algorithm and hardware. Within these simulation-based optimization problems the optimizer has to treat the simulation systems as black-boxes. The distributed solution of this kind of optimization problem demands efficient utilization of resources (i.e. processors) and evaluation of the solution quality. Analyzing the parallel performance is therefore an important task in the development of adequate distributed approaches taking into account the numerical algorithm, its implementation, and the used hardware architecture. In this paper, simulation-based optimization problems are characterized and a distributed solution algorithm is presented. Different performance analysis techniques (e.g. scalability analysis, computational complexity) are discussed and a new approach integrating parallel performance and solution quality is developed. This approach combines a priori and a posteriori techniques and can be applied in early stages of the solution process. The feasibility of the approach is demonstrated by applying it to three different classes of simulation-based optimization problems from groundwater management.     

Redundancies in Positive-Semidefinite Quadratic Programming

The question investigated is how to detect nonactive restrictions in positive-semidefinite quadratic programming. If the optimization problem satisfies some regularity conditions, we can use parametric optimization techniques for that analysis. It turns out that results obtained in Ref. 1, where only positive-definite matrices are considered, can be generalized to the semidefinite case. Simple calculations based exclusively on the problem data allow one to delete superfluous restrictions for this problem class during an optimization procedure.     

电力系统多目标优化运营研究

针对智能电网系统的安全与经济运行问题,建立了一个同时考虑经济、环境和安全指标的电网系统多目标优化模型,并运用理想点法对电网系统的多目标优化运营问题进行了相应的决策性分析,然后使用了一种新型的智能计算方法——标杆管理优化算法对该模型进行了求解计算.仿真实例表明,本文提出的决策分析和求解计算方法是切实可行的,具有一定的实用性和灵活性.此外,在计算过程中对一些相关的技术性问题,如对协调模型中的两类不同的控制变量、基因链的构造、约束条件的处理以及目标函数的选取等问题做了一些研究和探讨.     

Hahn-Banach theorems in nonstandard normed interval spaces

The conventional Hahn-Banach extension theorem based on vector space has been widely used to obtain many important and interesting results in nonlinear analysis, vector optimization and mathematical economics. Although the interval space is not a real vector space, the Hahn-Banach extension theorems based on interval spaces and nonstandard normed interval spaces can still be derived in this paper, which also shows the possible applications by considering the interval-valued problems in nonlinear analysis, vector optimization and mathematical economics.