箱约束变分不等式的一个简单光滑价值函数和阻尼牛顿法

通过引入中间值函数的一类光滑价值函数,构造了箱约束变分不等式的一种新的光滑价值函数,该函数形式简单且具有良好的微分性质．基于此给出了求解箱约束变分不等式的一种阻尼牛顿算法,在较弱的条件下,证明了算法的全局收敛性和局部超线性收敛率,以及对线性箱约束变分不等式的有限步收敛性．数值实验结果表明了算法可靠有效的实用性能．     

不等式组与变分不等式的极大熵函数方法

利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题，给出了不等式组及变分不等式问题近似解的可微优化方法，得到了不等式组和变分不等式问题的解集合的示性函数．     

运量分布与交通配流组合问题灵敏度分析方法

主要研究含单边约束的运量分布与交通配流组合问题的灵敏度分析计算方法.通过将该问题的数学规划模型等价转化为变分不等式模型,进而利用变分不等式问题的灵敏度分析方法,得到该组合问题中各决策变量关于扰动参数的导数公式.最后给出一个简单的数值算例说明该灵敏度分析方法的有效性.     

广义混合变分不等式的Levitin-Polyak适定性

首先给出广义混合变分不等式的Levitin-Polyak-α-近似序列以及适定性的定义.然后,定义广义混合变分不等式的gap函数并证明广义混合变分不等式的Levitin-Polyak适定性与其相应的gap函数的极小化问题的Levitin-Polyak适定性之间的等价性.最后,研究广义混合变分不等式的(广义)Levitin-Polyak-α-适定性的Furi-Vignoli型度量性质.     

变分不等式的极大熵函数方法

利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题,给出了不等式组及变分不等式问题近似解的可微优化方法,得到了不等式组和变分不等式问题的解集合的示性函数.     

广义向量变分不等式和广义对偶模型

研究实Banach空间中带有不等式约束的非光滑向量优化问题(VP).首先,借助下方向导数引进了广义Minty型向量变分不等式,并通过变分不等式来探讨问题(VP)的最优性条件.接着,利用函数的上次微分构造了不可微向量优化问题(VP)的广义对偶模型,并且在适当的弱凸性条件下建立了弱对偶定理.     

二阶锥约束随机变分不等式问题的数值方法研究

本文研究二阶锥约束随机变分不等式(SOCCSVI)问题,运用样本均值近似(SAA)方法结合光滑Fischer-Burmeister互补函数来求解该问题.首先,将SOCCSVI问题的Karush-Kuhn-Tucker系统转化为与之等价的方程组,并证明了该方程组的雅可比矩阵的非奇异性.其次,构造了光滑牛顿算法求解该方程组.最后,文章给出了两个数值实验证明了算法的有效性.     

有界约束单调变分不等式的内点连续算法

讨论变分不等式问题VIP(X，F)，其中F是单调函数，约束集X为有界区域．利用摄动技术和一类光滑互补函数将问题等价转化为序列合两个参数的非线性方程组，然后据此建立VIP(X，F)的一个内点连续算法．分析和论证了方程组解的存在性和惟一性等重要性质，证明了算法很好的整体收敛性，最后对算法进行了初步的数值试验。     

一类新的Lagrangian乘子法

本文提出了求解光滑不等式约束最优化问题新的乘子法,在增广Lagrangian函数中,使用了新的NCP函数的乘子法．该方法在增广Lagrangian函数和原问题之间存在很好的等价性;同时该方法具有全局收敛性,且在适当假设下,具有超线性收敛率．本文给出了一个有效选择参数C的方法．     

仿射变换内点信赖域方法求解变分不等式问题

本文研究了求解非线性约束变分不等式问题(VIP)的一个新的算法.利用KKT条件的非光滑方程形式,得到了与VIP等价的简单约束优化问题.提出了求解VIP的一类结合回代线搜索技巧的仿射变换内点信赖域算法.在较弱的条件下证明了算法具有整体收敛性,进一步在某些正则条件下,证明了算法具有超线性收敛速度.     

求解不可微箱约束变分不等式的下降算法

1 引 论 设X(?)Rn是非空闭集,F:Rn→Rn连续映射,变分不等式问题VI(X,F)是指:求x∈X,使 F(x)T(y-x)≥0,　 (?)y∈X,(1)记指标集N=(1,2,…,n},当 X=[a,b]≡{x∈Rn|a≤xi≤bi,i∈N},(2)其中a={a1,a2,…,an}T,b={b1,b2,…,bn}T∈Rn时,VI(X,F)化为箱约束变分不等式VI(a,b,F).若ai=0,bi=+∞,i∈N,即X=R+n≡{x∈Rn|x≥0}时,VI(a,b,F)化为非线性     

箱约束变分不等式的一种新NCP-函数及其广义牛顿法

１．引 言设，变分不等式，记为ＶＩ（Ｘ，Ｆ），是指：求ｘ＝Ｘ使记为箱式约束时，称 ＶＩ（Ｘ，Ｆ）为箱约束变分不等式，记为 ＶＩ（［ａ，ｂ］，Ｆ）．若ａｉ＝０，ｂｉ＝＋∞，　　　　　　　　　　　　　　　　　　　　　　　　　为非线性互补问题ＮＣＰ（Ｆ）：求ｘ∈Ｒ     

Solution of Finite-Dimensional Variational Inequalities Using Smooth Optimization with Simple Bounds

The variational inequality problem is reduced to an optimization problem with a differentiable objective function and simple bounds. Theoretical results are proved, relating stationary points of the minimization problem to solutions of the variational inequality problem. Perturbations of the original problem are studied and an algorithm that uses the smooth minimization approach for solving monotone problems is defined.     

盒子型区域上变分不等式的一个光滑牛顿法(英文)

The box constrained variational inequality problem can be reformulated as a nonsmooth equation by using median operator.In this paper,we present a smoothing Newton method for solving the box constrained variational inequality problem based on a new smoothing approximation function.The proposed algorithm is proved to be well defined and convergent globally under weaker conditions.     

MERIT FUNCTION AND GLOBAL ALGORITHM FOR BOX CONSTRAINED VARIATIONAL INEQUALITIES

1 IntroductionWe consider tlie variational inequality problelll, deuoted by VIP(X, F), wliicli is to find avector x* E X such thatF(X*)"(X -- X-) 2 0, VX E X, (1)where F: R" - R" is any vector-valued f11uction and X is a uonelllpty subset of R'.This problem has important applicatiolls. in equilibriun1 modeIs arising in fields such asecououtics, transportatioll scieuce alld operations research. See [1]. There exist mauy lllethodsfor solviug tlie variational li1equality problem VIP(X. …     

Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems

Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.