Generalized Connected Functions with Respect to Cones

In this paper, generalized connected functions with respect to cones such as quasi cone-connected, pseudo cone-connected, strongly pseudo cone-connected, and strictly pseudo cone-connected functions are introduced; necessary and sufficient optimality conditions are obtained for a weak minimum, a minimum, and a strong minimum of a vector-valued minimization problem. A Mond–Weir type dual is associated, and weak and strong duality results are established.     

Local-global properties of bifunctions

Representations of composite systems, such as bilinear programming, models of consumer/producer behavior, and sensitivity problems involve bifunctions (functions of two vector arguments). Such bifunctions are typically convex, pseudoconvex, or quasiconvex in each of their arguments, but not jointly convex, pseudoconvex, or quasiconvex. These functions do not in general possess the strong local-global property, namely, that every stationary point is a global minimum. In this paper, we define conditions that ensure that a bifunction possesses only a global minimum. In exploring this question, we use P-convexity and pseudo P-convexity, which are classes of bifunctions that generalize quasiconvexity and pseudoconvexity.     

对称弧式连通凸多目标半无限规划的最优性

以弧式连通函数和对称梯度为基础,研究新函数在多目标半无限规划下的最优性理论.定义了一类新的弧式连通函数,对称弧式连通函数、对称拟弧式连通函数、对称弱拟弧式连通函数、对称伪弧式连通函数、对称严格伪弧式连通函数,讨论了这些函数在多目标半无限规划下的最优性.给出更加广义的弧式连通函数,将它们运用到多目标半无限规划.     

一类广义凸函数及其等价刻画

引入了一个广义凸函数一高阶强Pre-invex函数,它是Guneer-Bhatia介绍的高阶强凸函数的一种推广.在此基础上,讨论并证明了高阶强Pre-invex函数的一些等价刻画,这些结论是Guneer-Bhatia给出结论的推广,扩大了凸函数的应用范围.     

Wellposed convex integral functionals

Sufficient conditions are obtained for wellposedness of convex minimum problems of the calculus of variations for multiple integrals under strong or weak perturbations of the boundary data. Problems with a unique minimizer as well as problems with several solutions are treated. Wellposedness under weak convergence of the boundary data in W^{1 p}

ω is proved if p

>2 and a counterexample is exhibited if p

=2.     

凸性与度量投影的连续性

本文研究近强凸、近非常凸Banach空间中度量投影的连续性。获得如下结果：若A是近强凸（近非常凸）空间中的逼近凸集,则度量投影PA是范-范上半连续的（范-弱上半连续的）。此外,我们还利用度量投影的连续性给出Banach空间为近强凸、近非常凸的一些充分必要条件。     

Weak and strong stationarity in generalized bilevel programming and bilevel optimal control

In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints (MPCC). We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented.     

非线性规划问题的二阶对称对偶性

本文讨论了二阶凸和二阶凹条件下的二阶对称对偶问题,并利用有效性和真有效性概念证明了弱对偶、强对偶、逆对偶及自对偶定理。     

Pseudo definably connected definable sets

In o‐minimal structures, every cell is definably connected and every definable set is a finite union of its definably connected components. In this note, we introduce pseudo definably connected definable sets in weakly o‐minimal structures having strong cell decomposition, and prove that every strong cell in those structures is pseudo definably connected. It follows that every definable set can be written as a finite union of its pseudo definably connected components. We also show that the projections of pseudo definably connected definable sets are pseudo definably connected. Finally, we compare pseudo definable connectedness with (recently introduced) weak definable connectedness of definable sets in weakly o‐minimal structures.     

New strong duality results for convex programs with separable constraints

It is known that convex programming problems with separable inequality constraints do not have duality gaps. However, strong duality may fail for these programs because the dual programs may not attain their maximum. In this paper, we establish conditions characterizing strong duality for convex programs with separable constraints. We also obtain a sub-differential formula characterizing strong duality for convex programs with separable constraints whenever the primal problems attain their minimum. Examples are given to illustrate our results.     

Exact penalty functions and stability in locally Lipschitz programming

In this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz; arbitrary simple constraints are also allowed. Assuming a weak stability condition, we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function. A tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied. We apply these results to unify and extend some results for convex programming. Since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions, our results provide a framework for extending these algorithms to problems with locally Lipschitz functions.     

Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data

In this paper, we establish some results for the increasing convex comparisons of generalized order statistics. First, we prove that if the minimum of two sets of generalized order statistics are ordered in the increasing convex order, then the remaining generalized order statistics are also ordered in the increasing convex order. This result is extended to the increasing directionally convex comparisons of random vectors of generalized order statistics. For establishing this general result, we first prove a new result in that two random vectors with a common conditionally increasing copula are ordered in the increasing directionally convex order if the marginals are ordered in the increasing convex order. This latter result is, of course, of interest in its own right.     

A modified wolfe dual for weak vector minimization

A version of the Wolfe dual problem is constructed for constained weak minimization of a vector objective function, in finite or infinite dimensions (e.g. continuous programming) The usual convex requirements are weakened to invex. Weak duality is replaced by an inclusion, constructed using the cone defining the weak minimum. Relations with Pareto (or proper Pareto) minima are discussed.     

Some Farkas-type results for fractional programming problems with DC functions

For a kind of fractional programming problem that the objective functions are the ratio of two DC (difference of convex) functions with finitely many convex constraints, in this paper, its dual problems are constructed, weak and strong duality assertions are given, and some sufficient and necessary optimality conditions which characterize their optimal solutions are obtained. Some recently obtained Farkas-type results for fractional programming problems that the objective functions are the ratio of a convex function to a concave function with finitely many convex constraints are the special cases of the general results of this paper.     

Optimality conditions for weak efficiency to vector optimization problems with composed convex functions

We consider a convex optimization problem with a vector valued function as objective function and convex cone inequality constraints. We suppose that each entry of the objective function is the composition of some convex functions. Our aim is to provide necessary and sufficient conditions for the weakly efficient solutions of this vector problem. Moreover, a multiobjective dual treatment is given and weak and strong duality assertions are proved.      

一类多目标Lipschitz规划的对偶定理

Lipschitz函数定义了广义本性伪凸的概念,建立了多目标Lipschitz规划的Mond-Weir型对偶和Wolfe型对偶,证明了原规划与对偶规划之间的对偶定理。     

Convergence of the Approximate Auxiliary Problem Method for Solving Generalized Variational Inequalities

We consider an extension of the auxiliary problem principle for solving a general variational inequality problem. This problem consists in finding a zero of the sum of two operators defined on a real Hilbert space H: the first is a monotone single-valued operator; the second is the subdifferential of a lower semicontinuous proper convex function . To make the subproblems easier to solve, we consider two kinds of lower approximations for the function : a smooth approximation and a piecewise linear convex approximation. We explain how to construct these approximations and we prove the weak convergence and the strong convergence of the sequence generated by the corresponding algorithms under a pseudo Dunn condition on the single-valued operator. Finally, we report some numerical experiences to illustrate the behavior of the two algorithms.     

Global weak sharp minima for convex (semi-)infinite optimization problems

We mainly consider global weak sharp minima for convex infinite and semi-infinite optimization problems (CIP). In terms of the normal cone, subdifferential and directional derivative, we provide several characterizations for (CIP) to have global weak sharp minimum property.     

Strong CHIP, normality, and linear regularity of convex sets

We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the property (G) introduced by Jameson. Normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property (strong CHIP) and various regularity properties. In particular, we prove that the weak normal property is a dual characterization of the strong CHIP, and the uniform normal property is a characterization of the linear regularity. Moreover, the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at is bounded away from 0 uniformly over all points in the intersection of these convex sets.

     

On the Mann-type iteration and the convex feasibility problem

The weak and strong convergence of a sequence generated by a Mann-type iteration are investigated in the frame of a real Hilbert space. Some applications to the projection method for the convex feasibility problem are given.