一类非光滑广义分式规划的Kuhn-Tucker型最优性必要条件

考虑一类非线性不等式约束的非光滑minimax分式规划问题;目标函数中的分子是可微函数与凸函数之和形式而分母是可微函数与凸函数之差形式,且约束函数是可微的.在Arrow- Hurwicz-Uzawa约束品性下,给出了这类规划的最优解的Kuhn-Tucker型必要条件.所得结果改进和推广了已有文献中的相应结果.     

一类非光滑多目标规划的K-T必要条件

本文对一类由可微函数与凸函数之和形式组成目标函数的多目标规划，分别在Kuhn-Tucker约束品性和Arrow—Hurwicz—Uzawa约束品性下，给出了其弱有效解的K—T必要条件，并给出了其特例(目标函数含||Bx||p的情形)的K—T必要条件，从而推广和改进了已有的结果。     

非光滑(F,ρ,)θ-d一致不变凸广义分式规划的对偶性

结合F-凸,η-不变凸及d一致不变凸的概念给出了非光滑广义(F,ρ,θ)-d一致不变凸函数;就一类在凸集C上目标函数为Lipschitz连续的带有可微不等式约束的广义分式规划,提出一个对偶,并利用在广义Kuhn-Tucker约束品性或广义Arrow-Hurwicz-Uzawa约束品性的条件下得到的最优性必要条件,证明相应的弱对偶定理、强对偶定理及严格逆对偶定理.     

(h,ч)-数学规划问题的必要条件(英文)

给出了(h,(?)-η伪凸函数的概念,利用Ben-Tal广义代数运算讨论了 它与η-伪凸函数之间的关系.当目标函数和约束函数均为(h,(?))-可微函数时,在广义 Slater约束规格下,得到了相应规划问题取得最优解的Kuhn-Tucker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.     

(h,ψ)-数学规划问题的必要条件

给出了（h,ψ）-η伪凸函数的概念，利用Ben-Tal广义代数运算讨论了它与η－伪凸函数之间的关系。当目标函数和约束函数均为（h,ψ）-可微函数时，在广义Slater约束规格下，得到了相应规划问题取得最优解的Kuhn-Tucker必要条件。     

求解多目标优化问题的邻近牛顿法

针对一类特殊的多目标优化问题,其每个目标函数为一个二阶连续可微凸函数与一个真凸但不必可微函数之和,提出了邻近牛顿法.我们引入了带线搜索的邻近牛顿法和不带线搜索的邻近牛顿法.在适当的条件下,我们证明了由这两类算法产生的序列的每个聚点是多目标优化问题的Pareto平稳点.此外,我们给出了它们在约束多目标优化和鲁棒多目标优化...     

一类线性约束下不可微最优化问题的可行下降方法

本文给出了一类线性约束下不可微量优化问题的可行下降方法，这类问题的目标函数是凸函数和可微函数的合成函数，算法通过解系列二次规划寻找可行下降方向，新的迭代点由不精确线搜索产生，在较弱的条件下，我们证明了算法的全局收敛性     

关于最大熵与信息熵之差的上界讨论

基于自信息函数为一可微凸函数这一事实,利用自信息函数所对应的Jenson离散型不等式,得到了有关最大熵与信息熵之差上界的一些结果.     

回收函数与函数的无界性

主要利用回收锥和回收函数来研究函数的下无界性。首先, 针对凸函数在非可微条件下,利用中值定理和回收锥刻画了凸函数次微分的性质, 并在此基础上给出了基于次可微条件下回收向量的充要条件。其次,将凸性推广到E-凸, 在一定条件下,利用回收函数研究了E-凸函数的下无界性。最后,通过举例说明这些结果不能推广到拟凸条件.     

On necessary conditions for a class of nondifferentiable minimax fractional programming

The Kuhn–Tucker-type necessary optimality conditions are given for the problem of minimizing a max fractional function, where the numerator of the function involved is the sum of a differentiable function and a convex function while the denominator is the difference of a differentiable function and a convex function, subject to a set of differentiable nonlinear inequalities on a convex subset C$C$ of Rⁿ${\mathbb{R}}^n$, under the conditions similar to the Kuhn–Tucker constraint qualification or the Arrow–Hurwicz–Uzawa constraint qualification or the Abadie constraint qualification. Relations with the calmness constraint qualification are given.     

Necessary optimality conditions for a class of nonsmooth vector optimization

The Kuhn-Tucker type necessary conditions of weak efficiency are given for the problem of minimizing a vector function whose each component is the sum of a differentiable function and a convex function, subject to a set of differentiable nonlinear inequalities on a convex subset C of ℝ ⁿ , under the conditions similar to the Abadie constraint qualification, or the Kuhn-Tucker constraint qualification, or the Arrow-Hurwicz-Uzawa constraint qualification. Supported by the National Natural Science Foundation of China (No. 70671064, No. 60673177), the Province Natural Science Foundation of Zhejiang (No.Y7080184) and the Education Department Foundation of Zhejiang Province (No. 20070306).     

Constraint qualifications in a class of nondifferentiable mathematical programming problems

The Kuhn-Tucker type necessary optimality conditions are given for the problem of minimizing the sum of a differentiable function and a convex function subject to a set of differentiable nonlinear inequalities on a convex subset C of , under the conditions similar to the Kuhn-Tucker constraint qualification or the Arrow-Hurwicz-Uzawa constraint qualification. The case when the set C is open (not necessarily convex) is shown to be a special one of our results, which helps us to improve some of the existing results in the literature.     

Second-order global optimality conditions for convex composite optimization

In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Constraint qualification in a general class of Lipschitzian mathematical programming problems

The Kuhn–Tucker type necessary optimality conditions are given for the problem of minimizing the sum of a differentiable function and a locally Lipschitzian function subject to a set of differentiable nonlinear inequalities on a convex subset C of , under the condition of a generalized Kuhn–Tucker constraint qualification or a generalized Arrow–Hurwicz–Uzawa constraint qualification. The case when the set C is open is shown to be a special one of our results, which helps us to improve some of the existing results in the literature. To finish we consider several test problems.     

Optimality conditions in convex optimization revisited

The phrase convex optimization refers to the minimization of a convex function over a convex set. However the feasible convex set need not be always described by convex inequalities. In this article we consider a convex feasible set which is described by inequality constraints that are locally Lipschitz and not necessarily convex or differentiable. We show that if the Slater constraint qualification and a simple non-degeneracy condition is satisfied then the Karush–Kuhn–Tucker type optimality condition is both necessary and sufficient.     

Simple bounds for solutions of monotone complementarity problems and convex programs

For a solvable monotone complementarity problem we show that each feasible point which is not a solution of the problem provides simple numerical bounds for some or all components of all solution vectors. Consequently for a solvable differentiable convex program each primal-dual feasible point which is not optimal provides simple bounds for some or all components of all primal-dual solution vectors. We also give an existence result and simple bounds for solutions of monotone compementarity problems satisfying a new, distributed constraint qualification. This result carries over to a simple existence and boundedness result for differentiable convex programs satisfying a similar constraint qualification.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based on work sponsored by National Science Foundation Grants MCS-8200632 and MCS-8102684.     

General constraint qualifications in nondifferentiable programming

We show that a familiar constraint qualification of differentiable programming has nonsmooth counterparts. As a result, necessary optimality conditions of Kuhn—Tucker type can be established for inequality-constrained mathematical programs involving functions not assumed to be differentiable, convex, or locally Lipschitzian. These optimality conditions reduce to the usual Karush—Kuhn—Tucker conditions in the differentiable case and sharpen previous results in the locally Lipschitzian case.     

Whitney differentiability of optimal-value functions for bound-constrained convex programming problems

ABSTRACT

In the spirit of the Whitney Extension Theorem, consider a function on a compact subset of Euclidean space to be ‘Whitney-differentiable’ if it is a restriction of a continuously Fréchet-differentiable function with an open domain. Whitney-differentiable functions have been shown to have useful (yet possibly nonunique) derivatives and calculus properties even on the boundaries of their domains. This article shows that optimal-value functions for bound-constrained convex programmes with Whitney-differentiable objective functions are themselves Whitney-differentiable, even when the linear-independence constraint qualification is not satisfied. This result extends classic sensitivity results for convex programmes, and generalizes recent work. As an application, sufficient conditions are presented for generating continuously differentiable convex underestimators of nonconvex functions for use in methods for deterministic global optimization in the multivariate McCormick framework. In particular, the main result is applied to generate Whitney-differentiable convex underestimators for quotients of functions with known Whitney-differentiable relaxations.