关于锥控制性质

设Y为R~p(p≥2)中非空子集,S为R~p中非平凡凸锥,由S在Y中可产生一个序关系“≥s”:即对任意t,u∈Y,若t-u∈S,则记t≥s~u。定义1 Y中的点y_0称为Y的有效点,若{y:y≥sy_0,y∈Y}=(y_0}。记Y中的有效点全体为E(Y|S)。由定义易知,E(Y|S)={y_0:(Y-y_0)∩S={0}}。     

准上半连续性不必蕴含上半连续性的例子

<正> 1.命 X,Y 是拓扑空间,多值映象 T:X→2~Y 称为上半连续的(upper semi-continuous),如果对任何 x_0∈X 和任何开集 G(?)T(x_0),存在 x_0 在 X 中的邻域 U(x_0)使得 x∈U(x_0)蕴含 T(x)(?)G.F.E.Browder 证明了下述卓越的不动点原理([1]定理3).定理1 命 K 是局部凸隔离实拓扑向量空间 E 的非空紧致凸集,T:K→2~E 上半连续,使得对每个 x∈K,T(x)(?)E 是非空闭凸集,命δ(K)={x∈K|(?)y∈E,使 x+λy(?)K,(?)λ>0}表示 K 的代数边界.假设对每个 x∈δ(K),存在 y∈K,z∈T(x)和λ>0使得z-x=λ(y-x),那么存在 x_0∈K 使 x_0∈T(x_0).     

关于K凹向量值函数的一类极值问题

本文讨论凸集的极值点与K凹向量值函数的一类极值问题之间的关系. 定义1 对于集合C中的点x,若有x=λy+(1-λ)z,其中0<λ<1,y,z∈C,就有x=y=z,则称x为C的极值点.C的所有极值点组成的集合记为extC. 定义2 设X,Y是实拓扑局部凸空间,Ω为X的非空紧凸子集,K为Y中的具有非     

限制步长法及其推广

本文给出一个新的限制步长算法并讨论了算法的收敛性质.考虑问题这里f(x)∈C~2,C是R~n中的闭凸集.对于给定集合K,实数h及点y,定义hK={hx|x∈K},y+K={y+x|x∈K},而(?)表示K之边界点集.1.限制步长算法算法Ⅰ任意取定     

向量极值问题的回归点解释

设 X 为欧氏空间 R~n,Y 为欧氏空间 R~m,g 为映 X 到 Y 的映射,A(?)X 是任意非空子集.在下述向量极值问题(VMP)(VMP) max g(x),s.t.x∈A中,K 是 Y 中非平凡闭凸锥,K≠{0},如果{x∈A|g(x)-g(x_0)∈K\{0}}=φ,则称 x_0∈A 为(VMP)的有效解;如果 intK≠φ,并且{x∈A|g(x)-g(x_0)∈intK)=φ,则称 x_0∈A 为(VMP)的弱有效解.     

有效极点的一个存在性定理

设 Y,∧ 均是 R~m 中的非空集合,称 x∈Y 为 Y 的一个有效点,如不存在 y∈Y,y≠x使 x∈y+∧.记 Y 的有效点集为 E(Y,∧).称 x∈Y 为 Y 的一个极点,如不存在 y∈Y,z∈Y,y≠z 使 x∈(y,z).记 Y 的极点集为 Y_e。记 Y 的有效极点集为 E_Y=Y_e∩E(Y,∧).Yu,L.P.在[1]中说明,若∧是凸锥,Y 是紧多面凸集,那么如果 E(Y,∧)≠φ,则Y 必有有效极点,即 E_Y≠φ.显然这个结论是线性多目标规划单纯形法的理论基础.     

空间微分方程奇点的指数

<正> 1.1设 D(?)R~3为有界连通开区域,其边界(?)D 为光滑封闭曲面.作光滑映射 f:(?)→R~3,当点 a∈R~3不在(?)D 的象 f((?)D)上时,有向的 f((?)D)绕点 a 的立体角与4π之商是整数,以 l(f,(?),a)表示,称为 f 绕点 a 的拓扑度.     

多值非线性系统最优控制的一阶必要条件

由 E,(?)的最大单调性,及 int(D(?))∩D(E)≠φ得,E+(?)为 Y→2~(Y′) 的最大单调算子.再由(H_(?))知 E+(?)为强制的,故由多值最大单调算子的满射性得,(1.3)存在唯一解 y∈D(E),即 y 满足(1.1).证毕.记θ为(1.1)从控制到状态 y 的解映射.现考虑其最优控制问题.设允许控制集(?)_(ad)为(?)=L~∞(0,T;U)中的有界弱~*     

圆周自映射的v极限集

设 f:s~1→s~1为连续映射。f 的回归点集和非游荡集分别记为 R 和Ω.xes~1,令v(x)=ω(x)∩α(x),其中ω(x)(α(x)为 x 的ω-(α-)极限集.令Γ=(?)v(x),若 y(?)s~1,记∧(y)=(?)ω(x).我们证明了:(1)Γ=∧(Ω)=∧(∧)=∧(Γ);(2)Ω-Γ是 s~1中无处稠密的可数集;(3)若以 x 为端点的每个开弧至少包含某个轨道中的的两点,则 x∈Γ;(4)若Γ-R≠φ,则Γ-R 为不可数集;(5)如(?)-R≠φ,则(?)-R 为无限集;(6)Γ=R 当且仅当(?)~(+)∩(?)~(-)=R.其中(?)~(+)((?)~(-))表示 R 的右(左)闭包。     

有向路和有向圈的控制集数

1引言设G=(V,E)为无向图．子集D (?)V(G)是无向图G的控制集,如果对于任意的y,∈V(G)-D,都存在x∈D,使xy∈E(G)．G的控制集D是G的分裂控制集,如果G中由V(G)-D导出的子图G〈V(G)-D〉是不连通的．G的一个控制集D是G的一个强(弱)控制集,若dG(x)≥d_G(y)(d_G(x)≤d_G(y)),其中d_G(x)表示G中与点x关联的边数．对于有向图H=(V,A),子集D(?)V(H)称为H的控制集,如果对于任意的y∈     

群体多目标最优化的Lagrange对偶性

本文建立了目标和约束为不对称的群体多目标最优化问题的Lagrange对偶规划，在问题的联合弱有效解意义下，得到群体多目标最优化Lagrange型的弱对偶定理、基本对偶定理、直接对偶定理和逆对偶定理。     

超有效意义下向量集值优化修整的Lagrange乘子型对偶

给出了一类加细的向量集值优化超有效解的最优性条件,由此给出了一种改进的Lagrange乘子型对偶,并建立了对偶的弱定理,正定理及逆定理。     

向量最优化问题的Lagrange对偶与择一定理

本文讨论无限维向量最优化问题的Lagrange对偶与弱对偶，建立了若干鞍点定理与弱鞍点定理．作为研究对偶问题的工具，建立了一个新的择一定理．     

Duality Theorems for Separable Convex Programming Without Qualifications

In the research of mathematical programming, duality theorems are essential and important elements. Recently, Lagrange duality theorems for separable convex programming have been studied. Tseng proves that there is no duality gap in Lagrange duality for separable convex programming without any qualifications. In other words, although the infimum value of the primal problem equals to the supremum value of the Lagrange dual problem, Lagrange multiplier does not always exist. Jeyakumar and Li prove that Lagrange multiplier always exists without any qualifications for separable sublinear programming. Furthermore, Jeyakumar and Li introduce a necessary and sufficient constraint qualification for Lagrange duality theorem for separable convex programming. However, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. In this paper, we study duality theorems for separable convex programming without any qualifications. We show that a separable convex inequality system always satisfies the closed cone constraint qualification for quasiconvex programming and investigate a Lagrange-type duality theorem for separable convex programming. In addition, we introduce a duality theorem and a necessary and sufficient optimality condition for a separable convex programming problem, whose constraints do not satisfy the Slater condition.     

Combination between global and local methods for solving an optimization problem over the efficient set

In this paper, we propose an algorithm for globally solving optimization problems over efficient sets. The algorithm is established based on a branch and bound scheme in which the bounding procedure is performed by using the well known weak duality theorem in Lagrange duality. A suitable combination of this algorithm with a local search procedure in d.c. optimization (named DCA) leads to a promising global algorithm, whose efficiency is more or less confirmed by computational experiments on a large set of test problems.     

两个泛函优化定理的证明

在泛函优化理论中,Lagrange乘子定理、对偶定理占有重要地位.建立了带有等式和不等式约束的泛函优化问题,并给出了广义Lagrange乘子定理、广义Lagrange对偶定理的证明.     

Quasi-relative interior-type constraint qualifications ensuring strong Lagrange duality for optimization problems with cone and affine constraints

In this article we provide weak sufficient strong duality conditions for a convex optimization problem with cone and affine constraints, stated in infinite dimensional spaces, and its Lagrange dual problem. Our results are given by using the notions of quasi-relative interior and quasi-interior for convex sets. The main strong duality theorem is accompanied by several stronger, yet easier to verify in practice, versions of it. As exemplification we treat a problem which is inspired from network equilibrium. Our results come as corrections and improvements to Daniele and Giuffré (2007) [9].     

Minimax Fractional Programming for <Emphasis Type="Italic">n</Emphasis>-Set Functions and Mixed-Type Duality under Generalized Invexity

We establish the sufficient optimality conditions for a minimax programming problem involving p fractional n-set functions under generalized invexity. Using incomplete Lagrange duality, we formulate a mixed-type dual problem which unifies the Wolfe type dual and Mond-Weir type dual in fractional n-set functions under generalized invexity. Furthermore, we establish three duality theorems: weak, strong, and strict converse duality theorem, and prove that the optimal values of the primal problem and the mixed-type dual problem have no duality gap under extra assumptions in the framework. This research was partly supported by the National Science Council, NSC 94-2115-M-033-003, Taiwan.     

Superefficiency in Vector Optimization with Nearly Subconvexlike Set-Valued Maps

In the framework of locally convex topological vector spaces, we establish a scalarization theorem, a Lagrange multiplier theorem and duality theorems for superefficiency in vector optimization involving nearly subconvexlike set-valued maps.     

On the optimality of some classes of invex problems

In this paper we define two notions: Kuhn–Tucker saddle point invex problem with inequality constraints and Mond–Weir weak duality invex one. We prove that a problem is Kuhn–Tucker saddle point invex if and only if every point, which satisfies Kuhn–Tucker optimality conditions forms together with the respective Lagrange multiplier a saddle point of the Lagrange function. We prove that a problem is Mond–Weir weak duality invex if and only if weak duality holds between the problem and its Mond–Weir dual one. Additionally, we obtain necessary and sufficient conditions, which ensure that strong duality holds between the problem with inequality constraints and its Wolfe dual. Connections with previously defined invexity notions are discussed.