关于线性多目标数学规划强有效解的讨论

文章[5]对多目标数学规划提出了一种新的关于局部强有效解、强有效解的概念,它是单目标规划中严格局部最优解、严格整体最优解概念的一种拓广,本文对线性多目标数学规划的强有效解作一些讨论。     

锥约束多目标规划的二阶对称对偶性质

给出一对锥约束多目标非线性规划的二阶对称对偶问题,以及二阶F凸函数类的概念.在二阶F凸假设下证明了真有效解的对偶性质———弱对偶性、强对偶性及逆对偶性.     

均衡约束为二阶锥约束广义方程的数学规划问题的二阶充分条件(英)

本文考虑一类均衡约束为二阶锥约束广义方程的数学规划问题. 我们通过一个非光滑映射的方向导数, 给出了临界锥的定义, 并建立它在可行点处的等价形式. 基于此临界锥, 我们提出了均衡约束为二阶锥约束广义方程的数学规划问题的二阶充分性条件, 并且验证了在适当的条件下, M-稳定点处的二阶充分性条件是二阶增长条件成立的充分条件.     

多目标数学规划的对偶性

<正> R.R.Egudo 和M.A.Hanson 在文[2]中讨论了如下一类多目标数学规划的对偶性其中f:R~n→R~k,g:R~n→R~m 是向量值函数,e=(1,1,…,1)~T ∈R~k,λ∈W~(++)={ω|ω_i>0,sum from i=1 to k ω_i=1}。文[2]对多目标非凸规划(VP)和(VD)关于真有效解给出了弱对偶和强对偶定理。本文将(VP)和(VD)推广为如下一类常闭凸锥约束的多目标数学规划问题     

包含约束下C^1，1多目标问题的二阶最优性条件

本文讨论当目标函数与支撑函数F是C1,2时,包含约束下多目标规划问题的二阶最优性条件,并根据向量函数的二阶次微分建立了有效解的二阶充分必要条件.     

有效解的刻划

对于多目标问题有效解的刻划,已有许多工作,在[3]中推广了[5]中单目标凸规划的极优解的 Fritz John 型必要条件,在 Slater 型条件假定下,进一步给出了多目标非可微凸规划有效解的必要条件(本文(7),(8)).[2]在假定 Slater 型条件成立时,证明类似于[3]的条件(本文(9),(10))可成为有效解的充要条件.考虑问题     

多目标规划的αk-较多有效解的几何特性

1引言 多目标规划问题有关解的几何特性是多目标规划理论研究中的重要课题,文[1]和[2]对多目标规划问题的锥有效解和锥弱有效解的基本性质进行了较系统的论述.     

多目标规划局部有效解的二阶条件

最优性条件的研究一直是多目标规划理论的一个热点,关于有效解的一阶最优性条件的研究,已有大量的文献涌现.可是关于有效解的二阶条件,其研究结果寥寥无几.分析其原因,恐怕主要有两方面.其一,绝大多数多目标优化方法还是基于先将问题标量化,然后借用线性规划或非线性规划中已有的一些成熟的方法来求解,这些方法中的一部分对二阶条件不作任何要求;其二,二阶条件的讨论需要更多的分析工具和更精致的分析     

多目标锥—广义凸规划有效解的充要条件

本文提出了ｎ维欧氏空间上的锥－凸、锥－伪凸、锥－拟凸向量值函数的概念，讨论了它们之间的关系，并在此基础上，对多目标数学规划问题关于凸锥∧有效解，撇开约束品性，讨论了它的充分必要条件。     

一类非光滑多目标规划问题的最优性条件

研究了一类非光滑多目标规划问题.这类多目标规划问题的目标函数为锥凸函数与可微函数之和,其约束条件是Euclidean空间中的锥约束.在满足广义Abadie约束规格下,利用广义Farkas引理和多目标函数标量化,给出了这一类多目标规划问题的锥弱有效解最优性必要条件.     

Second-order mollified derivatives and optimization

The class of strongly semicontinuous functions is considered. For these functions the notion of mollified derivatives, introduced by Ermoliev, Norkin and Wets, is extended to the second order. By means of a generalized Taylor’s formula, second order necessary conditions are proved for both unconstrained and constrained optimization.     

On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming

We study a multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point considered. We provide necessary second order optimality conditions and also sufficient conditions via a Fritz John type Lagrange multiplier rule and a set-valued second order directional derivative, in such a way that our sufficient conditions are close to the necessary conditions. Some consequences are obtained for parabolic directionally differentiable functions and C ^1,1 functions, in this last case, expressed by means of the second order Clarke subdifferential. Some illustrative examples are also given.     

A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations

In this paper, we consider a class of mathematical programs governed by second-order cone constrained parameterized generalized equations. We reformulate the necessary optimality conditions as a system of nonsmooth equations under linear independence constraint qualification and the strict complementarity condition. A set of second order sufficient conditions is proposed, which is proved to be sufficient for the second order growth of the stationary point. The smoothing Newton method in [40] is employed to solve the system of nonsmooth equations whose strongly BD-regularity at a solution point is demonstrated under the second order sufficient conditions. Several illustrative examples are provided and discussed.     

On the Hausdorff Solvability of Dirichlet Problem for a Class of Second Order Elliptic Systems

In this paper we study the Dirichlet problem for a class of second order linear elliptec complex equation and show that the problem is of Hausdorff type.We alse obtain the sufficient and necessary conditions for the solvability of the boundary value problem and the expression of the generalized solution.     

Local properties of algorithms for minimizing nonsmooth composite functions

This paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionF(x)=f(x)+h(c(x)) wheref andc are smooth buth(c) may be nonsmooth. Local convergence at a second order rate is established for the generalized Gauss—Newton method whenh is convex and globally Lipschitz and the minimizer is strongly unique. Local convergence at a second order rate is established for a generalized Newton method when the minimizer satisfies nondegeneracy, strict complementarity and second order sufficiency conditions. Assuming the minimizer satisfies these conditions, necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established. Necessary and sufficient conditions for a two-step superlinear rate of convergence are also established when only reduced curvature information is available. All these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems.This work was done while the author was a Research fellow at the Mathematical Sciences Research Centre, Australian National University.     

时间测度链上一类二阶动力方程的振动性

研究了时间测度链上的一类具有非线性中立项和变时滞的二阶非线性动力方程的振动性.通过引入参数函数和广义的Riccati变换,并借助时间测度链上的有关理论,得到了方程振动的几个充分条件.所得结果推广和改进了现有文献中相应的结果.     

Generalizations and Properties of the Principal Eigenvalue of Elliptic Operators in Unbounded Domains

Using three different notions of the generalized principal eigenvalue of linear second‐order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions for the Dirichlet problem. Relations between these principal eigenvalues, their simplicity, and several other properties are further discussed. © 2015 Wiley Periodicals, Inc.     

Extremal problems for discrete and differential inclusions of hyperbolic type with distributed parameters

For the first time, necessary and sufficient conditions for an extremum are proved for the first mixed problem for discrete and differential inclusions of hyperbolic type. Some of the results are generalized to the multidimensional case of a second order elliptic operator in bounded cylindrical domains.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1641–1649, December, 1990.     

Higher-order optimality conditions for strict local minima

In this work, we study a nonsmooth optimization problem with generalized inequality constraints and an arbitrary set constraint. We present necessary conditions for a point to be a strict local minimizer of order k in terms of higher-order (upper and lower) Studniarski derivatives and the contingent cone to the constraint set. In the same line, when the initial space is finite dimensional, we develop sufficient optimality conditions. We also provide sufficient conditions for minimizers of order k using the lower Studniarski derivative of the Lagrangian function. Particular interest is put for minimizers of order two, using now a special second order derivative which leads to the Fréchet derivative in the differentiable case.     

On generalized absolutely monotone functions

Sufficient conditions for generalized absolutely monotone functions to possess a Taylor-type expansion in terms of the corresponding Extended Tchebycheff systems were found by Karlin and Ziegler. The question of necessary conditions, however, was left open. In this paper we solve this question by finding necessary and sufficient conditions for the validity of the expansion. The structure of the cone of generalized absolutely monotone functions and its extreme rays are also discussed. The research of the second author was partially supported by U.S. Army Contract-DA-31-124-ARO-D-462 in the MRC, Madison, Wisconsin.