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.     

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.     

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.     

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.     

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.     

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...     

On Nonconvex Pseudomonotone Equilibrium Problems with Applications

Set-Valued and Variational Analysis - In this paper, we provide a further study for nonconvex pseudomonotone equilibrium problems and nonconvex mixed variational inequalities by using global...     

A Generalized Chordal Metric in Control Theory Making Strong Stabilizability a Robust Property

An abstract chordal metric is defined on linear control systems described by their transfer functions. Analogous to a previous result due to Partington (Linear Operators and Linear Systems. An Analytical Approach to Control Theory. London Mathematical Society Student Texts, vol. 60, Cambridge University Press, Cambridge, 2004) for $H^\infty $ , it is shown that strong stabilizability is a robust property in this metric.     

高维欧氏空间中的广义度量方程及其应用

本文利用代数的方法，证明了：对于两个等数量有限基本元素构成的集合，杨路和张景中关于高维欧氏空间E^n中的度量方程仍然成立，得到了一个广义度量方程，其特殊情况就是著名的Cayley定理．作为初步应用，给出了两个单形外接超球球心距和棱切超球球心距的两个公式.     

Perturbations of Generalized Mehler Semigroups and Applications to Stochastic Heat Equations with Levy Noise and Singular Drift

In this paper we solve the Kolmogorov equation and, as a consequence, the martingale problem corresponding to a stochastic differential equation of type dX _t =AX _t dt+b(X _t )dt+dY _t , on a Hilbert space E, where (Y _t ) _t0 is a Levy process on E,A generates a C ₀-semigroup on E and b:EE. Our main point is to allow unbounded A and also singular (in particular, non-continuous) b. Our approach is based on perturbation theory of C ₀-semigroups, which we apply to generalized Mehler semigroups considered on L ²(), where is their respective invariant measure. We apply our results, in particular, to stochastic heat equations with Levy noise and singular drift.     

广义度量空间和偏序集

广义度量空间和偏序集都具有函数空间．而函数空间的存在为数学构造和计算提供了很大方便．本文还讨论了广义度量空间和偏序集之间的相互转化问题．     

Periodic Solutions for Second-Order Differential Equations involving Nonconvex Superpotentials

The paper establishes the existence of a nonconstant periodic solution of a general second order nonautonomous Hamiltonian system with discontinuous nonlinearities. The multiplicity of solutions is also studied.     

一般无约束优化问题的广义拟牛顿法

对一般目标函数极小化问题的拟牛顿法及其全局收敛性的研究,已经成为拟牛顿法理论中最基本的开问题之一．本文对这个问题做了进一步的研究,对无约束优化问题提出一类新的广义拟牛顿算法,并结合Goldstein线搜索证明了算法对一般非凸目标函数极小化问题的全局收敛性．     

An Invariant-Point Theorem in Banach Space with Applications to Nonconvex Optimization

Journal of Optimization Theory and Applications - An invariant-point theorem and its equivalent formulation are established in which distance functions are replaced by minimal time functions. It is...