【作者】 孙菊贺；

【导师】 张立卫；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2008， 博士

【摘要】 变分不等式问题在运筹学、计算机科学、系统科学、工程技术、交通、经济与管理等许多方面有广泛应用。在二十世纪最后20年里,它受到许多学者的特别关注。另外,锥约束优化,尤其半定规划和二阶锥规划也是目前最优化领域的研究热点之一,而锥约束变分不等式的研究还很初步.本论文主要研究了锥约束变分不等式问题的数值方法的收敛性,包括求解二阶锥约束变分不等式的半光滑Newton方法与光滑函数方法,以及Hilbert空间中的变分不等式的迫近点算法的收敛性.本论文所阐述的主要研究结果可概括如下:1.第二章主要研究二阶锥约束变分不等式的半光滑Newton方法,运用Fischer-Burmeister函数将二阶锥约束变分不等式的Karush-Kuhn-Tucker条件转化为非光滑方程组问题Φ_FB=0,并给出了映射Φ_FB的Clark广义微分（?）Φ_FB的表达式。在一定条件下证明了该广义微分的非奇异性。提出采用Armijo线搜索的求解该非光滑方程组的修正牛顿法,证明了该算法的全局收敛性和局部超线性收敛性。最后给出数值例子验证了算法的有效性。2.第三章主要用光滑化方法求解二阶锥约束变分不等式。运用光滑化的Fischer-Burmeister函数将二阶锥约束变分不等式的Karush-Kuhn-Tucker条件转化为光滑方程组问题E=0.在一定条件下,证明了映射E的Jacobin矩阵JE的非奇异性。运用光滑的牛顿算法求解该光滑方程组问题,证明了算法的全局收敛性。最后给出数值例子验证了算法的有效性。3.第四章主要研究了基于预解算子理论的一般变分不等式的求解方法。我们引入了Hilbert空间中一类新的单调算子,即M-单调算子,建立了一般变分不等式和一个不动点问题的等价性。为了求解该不动点问题,本文提出了一个迫近点算法。在一定条件下证明了该迫近点算法的全局收敛性。而后,将上述理论应用到半定矩阵空间中的变分不等式的求解。为了保证算法的可行性,还另外给出了求解不动点问题的沂似解方法。更多还原

【Abstract】 It is well known that variational inequality problems have many important applications in operation research,computer science,system science,engineering technology,transportation, economics and management ect.In the last 20 years of the twentieth century,great attentions have been paid by many scholars in this direction.Meanwhile,conic optimization,especially semidefinite programming and second order cone programming,is an active field in optimization community.However,the study of variational inequality problems with cone constraints is not enough.Based on this observation,this dissertation focuses on the study of convergence analysis of numerical methods for variational inequality problems with cone constraints,including semismooth Newton methods and smoothing function methods for second-order cone constrained variational inequality problems and proximal point methods for variational inequality problems in Hilbert space.The main results of this dissertation can be summarized as follows:1.In the second chapter,the Karush-Kuhn-Tucker system of a second order cone constrained variational inequality problem is transformed into a semismooth system of equations with the help of Fischer-Burmeister operators over second order cones.The Clarke generalized differential of the semismooth mapping is presented.A modified Newton method with Armijo line search is proved to have global convergence with local superlinear rate of convergence under certain assumptions on the variational inequality problem.An illustrative example is given to show how the globally convergent method works.2.The third chapter mainly deals with the smooth method for second-order cone constrained variational inequality problems.The Karush-Kuhn-Tucker system of a second order cone constrained variational inequality problem is transformed into a smooth system of equations E=0 with the help of smoothing Fischer-Burmeister operators over second order cones.Under some conditions,we prove the Jacobin JE of E is nonsingular.A global convergent smooth Newton method is given for solving the smooth system of equations. 3.Chapter 4 discuss the method based on the resolvent operator for general variational inequality. A new monotonicity,M-monotonicity,is introduced.With the help of resolvent operator, an equivalence between the variational inequality VI(C,F+G) and the fixed point problem of a nonexpansive mapping is established.A proximal point algorithm is constructed to solve the fixed point problem,which is proved to have a global convergence under the condition that F in theⅥproblem is strongly monotone and Lipschitz continuous. Furthermore,a convergent path Newton method,which is based on the assumption that the projection mapping∏_C(·) is semismooth,is given for calculatingε-solutions to the sequence of fixed point problems,enabling the proximal point algorithm implementable.更多还原