Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space

We present two extensions of Korpelevich's extragradient method for solving the variational inequality problem (VIP) in Euclidean space. In the first extension, we replace the second orthogonal projection onto the feasible set of the VIP in Korpelevich's extragradient method with a specific subgradient projection. The second extension allows projections onto the members of an infinite sequence of subsets which epi-converges to the feasible set of the VIP. We show that in both extensions the convergence of the method is preserved and present directions for further research.     

Strong convergence of estimators as ε n -minimisers of optimisation problemsof optimisation problems

In the paper we prove strong consistency of estimators as solution of optimisation problems. The approach of the paper covers non-identifiable models, and models for dependent samples. We provide statements about consistency of M-estimators in regression models with random and with non-random design. The research was partially supported by the Deutsche Forschungsgemeinschaft (project number 436TSE113/40) and by the Grant Agency of the Czech Republic under Grant 201/03/1027.     

A stability result in quasi-convex programming

Combining a result on the lower semicontinuity of the intersection of two convex-valued multifunctions and the level set approach of epi-convergence, we obtain results on the epi-upper semicontinuity of the supremum and the sum of two families of quasi-convex functions. As a consequence, we give some condition ensuring the stability of a quasi-convex program under a perturbation of the objective functions and the constraint sets.     

Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibres

We describe the asymptotic behaviour of a cylindrical elasticbody, reinforced along identical -periodically distributed fibresof size r, with 0 < r < , filled in with some differentelastic material, when this small parameter goes to 0. Thecase of small deformations and small strains is considered.We exhibit a critical size of the fibres and a critical linkbetween the radius of the fibres and the size of the Lamécoefficients of the reinforcing elastic material. Epi-convergencearguments are used in order to prove this asymptotic behaviour.The proof is essentially based on the construction of appropriatetest functions.     

一类随机二层规划问题的近似求解方法收敛性分析

利用离散化的方法来处理连续型随机变量,将含有连续型随机变量的一类随机二层规划问题转化为一系列确定二层规划问题,证明了这种近似逼近问题的解上图收敛到原问题的解.     

半无限规划的一种离散逼近法

首先根据K-收敛理论及有限覆盖定理给出了半无限规划问题的一种新的离散方法；然后结合上图收敛性理论证明了半无限规划问题的一些新的收敛性命题；同时也指出了这种新的离散逼近方法与网格逼近法在收敛性及收敛速度上有一些相似的结论成立，但其在关于SIP问题收敛性和收敛结果的检验上比其他方法更为方便。