Metric Subregularity in Generalized Equations

In this article, we study the metric subregularity of generalized equations using a new tool of nonsmooth analysis. We obtain a sufficient condition for a generalized equation to be metrically subregular, which is not a necessary condition for metric regularity, using a subtle adjustment of the Mordukhovich coderivative. We apply these results to the study of the metric subregularity in a Cournot duopoly game.     

Metric Subregularity of Composition Set-Valued Mappings with Applications to Fixed Point Theory

In this paper we underline the importance of the parametric subregularity property of set-valued mappings, defined with respect to fixed sets. We show that this property appears naturally for some very simple mappings which play an important role in the theory of metric regularity. We prove a result concerning the preservation of metric subregularity at generalized compositions. Then we obtain, in purely metric setting, several fixed point assertions for set-valued mappings in local and global frameworks.     

非欧空间中的广义度量方程及其应用

本文改进了杨世国关于非欧空间中基本图形的度量方程,建立一个一般意义下的、应用更为方便的广义度量方程,作为其初步应用,导出了非欧空间中两个单形之间的一些有趣的几何关系．     

Metric Subregularity for a Multifunction

Metric subregularity is an important and active area in modern variational analysis and nonsmooth optimization. Many existing results on the metric subregularity were established in terms of coderivatives of the multifunctions concerned. This note tries to give a survey of the metric subregularity theory related to the coderivatives and normal cones.     

广义度量方程的改进及其应用(I)

众所周知, 度量方程作为距离几何的基本内容和工具之一,在几何约束求解中扮演着主要的角色.改进了杨定华关于$n$维欧氏空间中两个等数量有限基本元素构成集合的广义度量方程, 建立了更为一般意义的、应用方便的广义度量方程,作为其初步应用,导出了两个单形之间的一些有趣的矩阵恒等式关系.特别地,将其两边取行列式,可以简洁得到关于联系两个单形的几何恒等式.     

Quadratic Growth and Strong Metric Subregularity of the Subdifferential for a Class of Non-prox-regular Functions

Journal of Optimization Theory and Applications - This paper mainly studies the quadratic growth and the strong metric subregularity of the subdifferential of a function that can be represented as...     

G-凸度量空间中的广义KKM定理及其应用

在一类新的G-凸度量空间中建立了一类新的KKM定理,统一、改进和发展了文献中的相应结果.作为应用,得到了几个新的匹配定理和不动点定理.     

Local Existence of Strong Solutions to the Generalized MHD Equations

This paper devotes to consider the local existence of the strong solutions to the generalized MHD system with fractional dissipative terms $\Lambda^{2\alpha}u$ for the velocity field and $\Lambda^{2\alpha}b$ for the magnetic field, respectively. We construct the approximate solutions by the Fourier truncation method, and use energy method to obtain the local existence of strong solutions in $H^s(\mathbb{R}^n)\,(s>\max \left\{\frac{n}{2}+1-2\alpha, 0\right\})$ for any $\alpha\geq0$.     

Global Existence of Strong Solutions for the Generalized Navier-Stokes Equations with Damping

Acta Mathematicae Applicatae Sinica, English Series - This paper mainly focus on the global existence of the strong solutions for the generalized Navier-Stokes equations with damping. We obtain the...     

Strong Solutions of Semilinear Parabolic Equations with Measure Data and Generalized Backward Stochastic Differential Equations

We prove that under natural assumptions on the data strong solutions in Sobolev spaces of semilinear parabolic equations in divergence form involving measure on the right-hand side may be represented by solutions of some generalized backward stochastic differential equations. As an application we provide stochastic representation of strong solutions of the obstacle problem by means of solutions of some reflected backward stochastic differential equations. To prove the latter result we use a stochastic homographic approximation for solutions of the reflected backward equation. The approximation may be viewed as a stochastic analogue of the homographic approximation for solutions to the obstacle problem.     

On Directional Metric Regularity, Subregularity and Optimality Conditions for Nonsmooth Mathematical Programs

This paper mainly deals with the study of directional versions of metric regularity and metric subregularity for general set-valued mappings between infinite-dimensional spaces. Using advanced techniques of variational analysis and generalized differentiation, we derive necessary and sufficient conditions, which extend even the known results for the conventional metric regularity. Finally, these results are applied to non-smooth optimization problems. We show that that at a locally optimal solution M-stationarity conditions are fulfilled if the constraint mapping is subregular with respect to one critical direction and that for every critical direction a M-stationarity condition, possibly with different multipliers, is fulfilled.     

Quadratic Majorization for Nonconvex Loss with Applications to the Boosting Algorithm

Classical robust statistical methods dealing with noisy data are often based on modifications of convex loss functions. In recent years, nonconvex loss-based robust methods have been increasingly popular. A nonconvex loss can provide robust estimation for data contaminated with outliers. The significant challenge is that a nonconvex loss can be numerically difficult to optimize. This article proposes quadratic majorization algorithm for nonconvex (QManc) loss. The QManc can decompose a nonconvex loss into a sequence of simpler optimization problems. Subsequently, the QManc is applied to a powerful machine learning algorithm: quadratic majorization boosting algorithm (QMBA). We develop QMBA for robust classification (binary and multi-category) and regression. In high-dimensional cancer genetics data and simulations, the QMBA is comparable with convex loss-based boosting algorithms for clean data, and outperforms the latter for data contaminated with outliers. The QMBA is also superior to boosting when directly implemented to optimize nonconvex loss functions. Supplementary material for this article is available online.     

BCQ条件和广义限制域最佳一致逼近的特征

研究了广义限制域的最佳一致逼近问题,在允许有有限个节点的情况下,引入次强内点条件的概念,并将优化理论中的BCQ条件等概念应用到本文所研究的问题中,刻划了次强内点条件、BCQ条件和最佳一致逼近的特征之间的关系.     

广义模糊Volterra积分方程与强模糊Henstock积分

研究形如的模糊Volterra积分方程整体解的存在性.这里h,k,g是实值函数,f为强模糊Henstock可积的模糊数值函数.所用的方法和工具是利用模糊数值函数的等度可积性和非紧性测度的性质以及广义Darbo不动点定理.     

Linear Convergence of Prox-SVRG Method for Separable Non-smooth Convex Optimization Problems under Bounded Metric Subregularity

Journal of Optimization Theory and Applications - With the help of bounded metric subregularity which is weaker than strong convexity, we show the linear convergence of proximal stochastic...     

Globally Convergent Variable Metric Method for Nonconvex Nondifferentiable Unconstrained Minimization

A special variable metric method is given for finding the stationary points of locally Lipschitz continuous functions which are not necessarily convex or differentiable. Time consuming quadratic programming subproblems do not need to be solved. Global convergence of the method is established. Some encouraging numerical experience is reported.     

模糊度量空间的强嵌入北大核心CSCD

本文定义了George和Veeramani意义下的模糊度量空间的强嵌入,证明了可强嵌入的模糊度量空间能够粗嵌入到Hilbert空间.另外还证明了强嵌入在模糊度量空间的粗范畴下是不变的,并给出了模糊度量空间强嵌入的一些等价刻画.     

The Strong Solution of a Class of Generalized Navier-Stokes Equations

We study initial boundary value (lBV) problem for a class of generalized Navier-Stokes equations in L^q([0, T); L^p(Ω)). Our main tools are regularity of analytic semigroup by Stokes operator and space-time estimates. As an application we can obtain some classical results of the Navier-Stokes equations such as global classical solution of 2-dimensional Navier-Stokes equation etc.