一个择一定理及对广义凸规划的应用

本文在实线性空间中得出了广义凸函数的择一定理 ,利用这一定理 ,我们获得广义凸规划的最优性条件     

一类非光滑规划问题的最优性和对偶

研究一类非光滑多目标规划问题,给出了该规划问题的三个最优性充分条件.同时,研究了该问题的对偶问题,给出了相应的弱对偶定理和强对偶定理.     

集合函数多目标规划的一阶最优性条件

在文（１）－（４）的基础上，本文通过引入集团函数的伪凸，严格伪凸，拟凸，严格拟凸等新概念，给出了集合函数多目标规划问题有效解的一阶充分条件，弱有效解的阶必要条件，弱有交解的一阶必要条件以及强有效解的一阶充分条件。     

Invex nonsmooth alternative theorem and applications *

An invex constrained nonsmooth optimization problem is considered, in which the presence of an abstract constraint set is possibly allowed. Necessary and sufficient conditions of optimality are provided and weak and strong duality results established. Following Geoffrion’s approach an invex nonsmooth alternative theorem of Gordan type is then derived. Subsequently, some applications on multiobjective programming are then pursued     

New Sufficiency for Global Optimality and Duality of Mathematical Programming Problems via Underestimators

We present new conditions for a Karush-Kuhn-Tucker point to be a global minimizer of a mathematical programming problem which may have many local minimizers that are not global. The new conditions make use of underestimators of the Lagrangian at the Karush-Kuhn-Tucker point. We establish that a Karush-Kuhn-Tucker point is a global minimizer if the Lagrangian admits an underestimator, which is convex or, more generally, has the property that every stationary point is a global minimizer. In particular, we obtain sufficient conditions by using the fact that the biconjugate function of the Lagrangian is a convex underestimator at a point whenever it coincides with the Lagrangian at that point. We present also sufficient conditions for weak and strong duality results in terms of underestimators. The authors are grateful to Professor Gue Myung Lee, Pukyong National University, Korea, and the referees for their comments and suggestions which have contributed to the final preparation of the paper. The work was partially supported by the Australian Research Council Discovery Project Grant.     

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].     

On regularity of a weak solution to the Navier–Stokes equation with generalized impermeability boundary conditions

We consider the 3D Navier–Stokes equation with generalized impermeability boundary conditions. As auxiliary results, we prove the local in time existence of a strong solution (‘strong’ in a limited sense) and a theorem on structure. Then, taking advantage of the boundary conditions, we formulate sufficient conditions for regularity up to the boundary of a weak solution by means of requirements on one of the eigenvalues of the rate of deformation tensor. Finally, we apply these general results to the case of an axially symmetric flow with zero angular velocity.     

Nonsmooth minimax fractional programming involving η-pseudolinear functions

This article deals with the necessary and sufficient optimality conditions for a class of nonsmooth minimax fractional programming problems with locally Lipschitz η-pseudolinear functions. Utilizing these optimality criteria, we formulate two types of dual models and establish weak and strong duality results. The results of this article extend several known results from the literature to a wider class of optimization problems.     

Some characterizations of duality for DC optimization with composite functions

In this paper, by virtue of the epigraph technique, we first introduce some new regularity conditions and then obtain some complete characterizations of the Fenchel–Lagrange duality and the stable Fenchel–Lagrange duality for a new class of DC optimization involving a composite function. Moreover, we apply the strong and stable strong duality results to obtain some extended (stable) Farkas lemmas and (stable) alternative type theorems for this DC optimization problem. As applications, we obtain the corresponding results for a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator, respectively.     

A Stone-Weierstrass theorem for Banach function spaces satisfying a certain separation property

We consider a strong lattice property for a Banach function space B on a compact Hausdorff space, which gives a general Stone-Weierstrass theorem for B. We also study the relation of this theorem and its proof to a certain decomposition of an associated compactification, and to another lattice-like property.     

序线性空间中向量极值问题的最优性条件

本文在序线性空间中建立了广义次似凸映射下的择一定理,运用此定理,得出一类向量极值问题的最优性条件.     

Theorems of the Alternative and Optimization with Set-Valued Maps

In this paper, the concept of generalized cone subconvexlike set-valued mapsis presented and a theorem of alternative for the system of generalizedinequality–equality set-valued maps is established. By applying thetheorem of the alternative and other results, necessary and sufficientoptimality conditions for vector optimization problems with generalizedcone subconvexlike set-valued maps are obtained.     

Constraint Qualifications in Nonsmooth Multiobjective Optimization

For an inequality constrained nonsmooth multiobjective optimization problem involving locally Lipschitz functions, stronger KT-type necessary conditions and KT necessary conditions (which in the continuously differentiable case reduce respectively to the stronger KT conditions studied recently by Maeda and the usual KT conditions) are derived for efficiency and weak efficiency under several constraint qualifications. Stimulated by the stronger KT-type conditions, the notion of core of the convex hull of the union of finitely many convex sets is introduced. As main tool in the derivation of the necessary conditions, a theorem of the alternatives and a core separation theorem are also developed which are respectively extensions of the Motzkin transposition theorem and the Tucker theorem.     

非凸向量集值优化Benson真有效解的最优性条件与对偶

在无需偏序锥内部非空的情况下给出了非凸约束向量集值优化Benaon真有效解一种加细的最优性条件，并建立了向量集值优化Benson真有效解一种改进的Lagrange乘子型对偶，它比已有的Lagrange乘子型对偶具有较好的对偶性。     

一类向量极值问题的最优性条件

本文利用序线性空间中关于次似凸集值映射的择一性定理,得出了具有广义等式和不等式约束的向量极值问题的最优性条件.     

弧连通锥-凸数学规划的最优性条件

在弧连通锥-凸假设下讨论Hausdorff局部凸空间中的一类数学规划的最优性条件问题.首先,利用择一定理得到了锥约束标量优化问题的一个必要最优性条件.其次,利用凸集分离定理证明了无约束向量优化问题关于弱极小元的标量化定理和一个一致的充分必要条件.所得结果深化和丰富了最优化理论及其应用的内容.     

向量极值问题的一种含有梯度的最优性条件

本文在赋范空间中引入 G-可微函数的梯度概念 ,利用择一定理 ,我们获得了向量极值问题含有梯度的的最优性条件     

LMI优化问题的择一性定理与对偶

考虑目标函数是线性函数约束条件为线性矩阵不等式的LMI优化问题,讨论了LMI优化问题中的四个择一性定理.每种类型的择一性定理包含两个线性不等式和(或)等式系统,一个原始系统和一个对偶系统.弱择一性定理说明两系统中至多只有其一有解;基于凸集分离理论得到的强择一性定理说明两系统有且仅有其一有解.并在此基础上推导了LMI优化...     

The weak capacity of averaged channels

Summary Coding theorem and weak converse of the coding theorem are proved for averaged semicontinuous stationary channels and for almost periodic discrete channels, whose phases are statistically known. Explicit formulas for the capacities are given. The strong converses of the coding theorems do not hold.     

(u,02)-广义次似凸映射的择一定理及其对向量优化的应用

在序线性拓扑空间中,引入（u,02）-广义次似凸映射,建立了（u,02）-广义次似凸映射的一个择一定理.最后,利用此定理,在序线性拓扑空间中得到了带广义不等式约束的向量极值问题的最优性必要条件和充分条件.