Banach空间中的相补问题

本文在Ｂａｎａｃｈ空间中研究了三类相补问题解的存在性。所得结果是［４，５，６，９，１１－１４］中相应结果的深入和发展。     

New smooth C-functions for symmetric cone complementarity problems

In this paper, with the help of the Jordan-algebraic technique we introduce two new complementarity functions (C-functions) for symmetric cone complementary problems, and show that they are continuously differentiable and strongly semismooth everywhere. The work was partly supported by the National Natural Science Foundation of China (10671010, 70471002).     

一个半正Sturm-Liouville边值问题的n个解的存在性

通过构造与非线性项有关的辅助函数并考察这个辅助函数在有界集上的性质，对于Sturm-Liouville边值问题建立了几个解的存在性，其中非线性项是下有界的．同时，给出了正解和非平凡解的一个判别方法．主要工具是锥压缩与锥拉伸型的Krasnosel’skii不动点定理．     

Implicit function theorem without a priori assumptions about normality

The equation F(x, σ) = 0,x ∈ K, in which σ is a parameter and x is an unknown taking values in a given convex cone in a Banach space X, is considered. This equation is examined in a neighborhood of a given solution (x _*, σ_*) for which the Robinson regularity condition may be violated. Under the assumption that the 2-regularity condition (defined in the paper), which is much weaker than the Robinson regularity condition, is satisfied, an implicit function theorem is obtained for this equation. This result is a generalization of the known implicit function theorems even for the case when the cone K coincides with the entire space X.     

A new smoothing Newton-type method for second-order cone programming problems

A new smoothing function of the well-known Fischer–Burmeister function is given. Based on this new function, a smoothing Newton-type method is proposed for solving second-order cone programming. At each iteration, the proposed algorithm solves only one system of linear equations and performs only one line search. This algorithm can start from an arbitrary point and it is Q-quadratically convergent under a mild assumption. Numerical results demonstrate the effectiveness of the algorithm.     

关于随机算子不动点指数的几个定理

研究了随机半闭1-集压缩算子和随机凝聚算子的随机不动点指数问题,推广了郭大钧文中的几个定理.     

测度链上非线性m-点边值问题的正解

设T是一个测度链(时间标架)并且0,T∈T．讨论测度链上m-点边值问题其中a∈Cld((0,T),[0,∞)),f∈C((0,T)X[0,∞),[0,∞)),β,γ∈[0,∞),ξ∈(0,ρ(T)), ai∈[0,∞)(i=1,．．．,m-2)是一些满足适当条件的定常数．借助于锥上的不动点定理,得到了此问题存在单个及多个正解的一些新的更一般的结果．特别地,我们的结果推广并改进了一些已有的结论．     

Quadratic cone cutting surfaces for quadratic programs with on–off constraints

We study the convex hull of a set arising as a relaxation of difficult convex mixed integer quadratic programs (MIQP). We characterize the extreme points of the convex hull of the set and the extreme points of its continuous relaxation. We derive four quadratic cutting surfaces that improve the strength of the continuous relaxation. Each of the cutting surfaces is second-order-cone representable. Via a shooting experiment, we provide empirical evidence as to the importance of each inequality type in improving the relaxation. Computational results that employ the new cutting surfaces to strengthen the relaxation for MIQPs arising from portfolio optimization applications are promising.     

A complete characterization of disjunctive conic cuts for mixed integer second order cone optimization

We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization (MISOCO) problem. We extend our prior work on disjunctive conic cuts (DCCs), which has thus far been restricted to the case in which the intersection of the hyperplanes and the feasible set is bounded. Using a similar technique, we show that one can extend our previous results to the case in which that intersection is unbounded. We provide a complete characterization in closed form of the conic inequalities required to describe the convex hull when the hyperplanes defining the disjunction are parallel.     

Exact separation theorem for closed sets in Asplund spaces

In this paper, in terms of the Fréchet normal cone, we establish exact separation results for finitely many disjoint closed sets in an Asplund space, which supplement the extremal principle and some fuzzy separation theorems. As an application, we provide a new optimality condition for a constraint optimization problem in terms of Fréchet subdifferential and Fréchet normal cone.