半局部凸多目标规划的最优性条件与对偶定理

得到了半局部凸多目标规划．G-真有效解的充要条件，建立了Mond-Weir型对偶模型,证明了相应的对偶定理。     

一类非凸集值优化问题的对偶定理

对偶理论是数学规划研究领域的重点问题之一,通对偶模型可以实现一个最小化问题与一个最大化问题之间的相互转化.本文的目的是建立一类非凸约束集值优化问题的对偶理论,在逼近多值函数定义的不变凸性假设下,研究了原集值优化问题的Mond-Weir型和Wolfe型对偶问题.利用分析的方法,本文得到了两种对偶模型下关于弱极小元的弱对偶定理,强对偶定理和逆对偶定理.这些对偶定理揭示了原问题与所讨论的Mond-Weir型和Wolfe型对偶问题之间存在着明确的对偶关系.本文所得结果丰富和深化了集值优化理论及其应用的研究内容.     

一类非凸集值优化问题的对偶定理（英文）

对偶理论是数学规划研究领域的重点问题之一,通对偶模型可以实现一个最小化问题与一个最大化问题之间的相互转化.本文的目的是建立一类非凸约束集值优化问题的对偶理论,在逼近多值函数定义的不变凸性假设下,研究了原集值优化问题的Mond-Weir型和Wolfe型对偶问题.利用分析的方法,本文得到了两种对偶模型下关于弱极小元的弱对偶定理,强对偶定理和逆对偶定理.这些对偶定理揭示了原问题与所讨论的Mond-Weir型和Wolfe型对偶问题之间存在着明确的对偶关系.本文所得结果丰富和深化了集值优化理论及其应用的研究内容.     

厚板哈密顿求解体系及其变分原理与正交关系

将哈密顿求解体系推广应用于Reissner-Mindlin厚板问题.首先导出了厚板哈密顿对偶微分方程,然后采用换元乘子法导出了厚板哈密顿变分原理的泛函表示式,最后提出并证明了厚板理论的两个正交关系.厚板哈密顿体系的理论成果将为研究厚板解析解和有限元解提供新的有效工具.     

基于对偶理论的椭圆变分不等式的后验误差分析（英文）

本文基于对偶理论对椭圆变分不等式的正则化方法提供一个相对全面的后验误差分析.我们分别考虑了摩擦接触问题和障碍问题,通过选取一种不同形式的有界算子和泛函,推导了其对偶形式并给出了正则化方法的H~1范后验误差估计.最后,利用凸分析中的对偶理论建立了障碍问题的残量型后验误差估计的一般框架.同时我们选取一种特殊的对偶变量和泛函形式得到该问题的残量型误差估计及其有效性.数值解的后验误差估计是发展有效自适应算法的基础而模型误差的后验误差估计在分析问题中数据的不确定影响时是非常有用的.     

非光滑多目标分式规划的对偶性

利用方向导数对非光滑函数引入了伪凸、严格伪凸和扭凸等概念，给出了非光滑多目标分式规划的对偶同题，井证明了弱对偶定理、强对偶定理及严格对偶定理．     

D.C.集(凸集的差)约束的非凸二次规划的最优解集

本文研究D.C.集(凸集的差)上极小化非凸二次规划问题的最优解。我们首先证明了该问题的Lagrange对偶的稳定性,即不存在对偶间隙;接着利用该性质得到问题的全局最优性条件和最优解集,它可以像凸规划那样,借助它的对偶问题的解集精确地描述出来。最后,通过一个例子来说明这些结论。     

基于序列线性组合的原始–对偶算法

双线性鞍点问题及其对应的原问题和对偶问题在信号图像处理、机器学习、统计和高维数据处理等领域具有重要的应用,原始对偶算法是求解该类问题的有效算法。利用序列的线性组合技术,改进了Chambolle-Pock原始对偶算法子问题的求解,提出了一种求解双线性鞍点问题的新原始对偶算法。该算法也是Arrow-Hurwicz算法的修正,在子问题求解中将线性组合和经典的外插技术进行结合,得到了更一般的收敛性。利用变分分析证明了算法的收敛性和遍历■(1/N)收敛率,获得了保证算法收敛的步长和组合参数取值范围,求解非负最小二乘和Lasso问题的数值实验验证了算法的有效性。     

Banach空间中强增生型变分包含解的迭代逼近问题

本文研究p-一致光滑的实Banach空间（1＜p≤2）中一类强增生型变分包含解的迭代逼近问题.在仅假设强增生映象的连续性与没有条件βn→0（n→∞）下,利用徐宗本等人给出的对偶映象Jp的Holder连续性,证明了具误差的Ishikawa迭代程序强收敛到这类变分包含的唯一解.     

具有双工期的最小化最大延迟的双目标排序

本文研究一个双目标排序问题.由于决策者的利益不同,他们对工件的加工有不同的工期要求,因而可导出两个最大误工问题.我们证明了可在多项式时间内找到关于这两个目标函数的所有Pareto最优点.     

Duality in fuzzy linear programming: a survey

The concepts of both duality and fuzzy uncertainty in linear programming have been theoretically analyzed and comprehensively and practically applied in an abundance of cases. Consequently, their joint application is highly appealing for both scholars and practitioners. However, the literature contributions on duality in fuzzy linear programming (FLP) are neither complete nor consistent. For example, there are no consistent concepts of weak duality and strong duality. The contributions of this survey are (1) to provide the first comprehensive overview of literature results on duality in FLP, (2) to analyze these results in terms of research gaps in FLP duality theory, and (3) to show avenues for further research. We systematically analyze duality in fuzzy linear programming along potential fuzzifications of linear programs (fuzzy classes) and along fuzzy order operators. Our results show that research on FLP duality is fragmented along both dimensions; more specifically, duality approaches and related results vary in terms of homogeneity, completeness, consistency with crisp duality, and complexity. Fuzzy linear programming is still far away from a unifying theory as we know it from crisp linear programming. We suggest further research directions, including the suggestion of comprehensive duality theories for specific fuzzy classes while dispensing with restrictive mathematical assumptions, the development of consistent duality theories for specific fuzzy order operators, and the proposition of a unifying fuzzy duality theory.     

集值优化问题严有效意义下的鞍点与对偶

本文研究集值优化问题在严有效意义下的鞍点理论及对偶理论。首先,给出集值优化问题在严有效意义下鞍点的一个等价命题和鞍点定理。其次,在某种凸性假设下,研究严有效意义下鞍点最优性条件、弱对偶性、强对偶性、逆对偶性。     

A duality theorem for plastic plates

Summary Limit analysis studies the asymptotic behavior of elastic-plastic materials and structures. The asymptotic material properties exist for a class of ductile metals and are designed into optimal structural members such as I-beams and composite plates. The analysis automatically ignores the relatively small elastic deformations. Classical lower and upper bound theorems in the form of inequalities are mathematically incomplete. A duality theorem equates the greatest lower bound and the least upper bound. Although some general statement has been made on the duality relation of limit analysis, each yield criterion will lead to a specific duality theorem. The duality theorem for a class of plastic plates is established in this paper. The family of -norms is used to represent the yield functions. Exact solutions for circular plates under a uniform load are obtained for clamped and simply supported boundaries as examples of the specific duality relations. Two classical solutions associated with Tresca and Johansen yield functions are also presented in the spirit of their own duality relations, providing interesting comparison to the new solutions. A class of approximate solutions by a finite element method is presented to show the rapid mesh convergence property of the dual formulation. Complete and general forms of the primal and dual limit analysis problems for the -family plates are stated in terms of the components of the moment and curvature matrices.With 6 Figures     

Dynamic financial planning: certainty equivalents, stochastic constraints and functional conjugate duality

This paper studies portfolios under risk and stochastic constraints. Certainty equivalents combine risk aversion and exponential utility to form the objective. Budget and stochastic constraints on the account balance are used to ensure a positive net worth over time. These portfolio models are analyzed by functional conjugate duality for general distributions and by conjugate duality for the normal distribution. All the programs are convex. The duals provide insight into this approach and relate it to other stochastic and financial concepts.     

非线性规划的二阶对称对偶性

这篇文章提出了一对二阶对称对偶模型并建立了对偶性定理．     

一阶半线性双曲型方程组的精确边界能控性与能观性

本文以一阶半线性双曲型方程组混合初边值问题的整体C~1解理论为基础,采用直接构造的方法建立了一阶半线性双曲组的整体精确边界能控性及能观性理论,对于相应的控制及观测时间给出了精确的估计,并揭示了精确能控性与能观性之间隐含的某种对偶关系。本文还揭示了在非自治系统的情形下,精确能控性与能观性的对偶关系有可能会丧失。     

Surrogate Duality Based Method for Contact Problems

We present a numerical method based on surrogate duality to solve contact problems in elasticity. The primal optimization problem arising from the contact problem is converted to a corresponding dual problem using surrogate duality. An explicit dual formulation for the surrogate dual problem is obtained on some certain conditions, and a Karmarkar's interior point based method is introduced to solve the explicit surrogate dual problem. Numerical examples including some quadratic programming problems with dense matrices and one elastic contact problem are solved using the present method to show its feasibility and efficiency.     

Stripe-Superconductor Duality and the Emergence of Nodal Fermions

Recent experiments suggest a stripe-superconductor duality to be operating for some regime of the x vs. T phase diagram of the cuprate oxide superconductors. Extending a nonperturbative approach previously applied to quasi-1d systems, we formulate this duality where special emphasis is laid on the role of global antiferromagnetic fluctuations which are enhanced in the vicinity of stripe dislocations. A condensate of such antiferromagnetic fluctuation is found to be able to support nodal fermions.