【作者】 刘波；

【导师】 张立卫；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2006， 硕士

【摘要】 本文主要研究求解非线性互补问题的微分方程方法,包括求解非线性互补问题的一阶微分方程方法和二阶微分方程方法的理论及相应的数值实现。 互补问题是数学规划中的一个非常重要的分支。它为研究线性和二次规划提供了一个普遍框架。它与不动点理论,变分不等式问题,线性和非线性分析,以及其他领域的应用数学如经济,平衡问题等都有密切的联系。求解非线性互补问题有许多有效的算法,如不动点方法,同伦算法,投影方法,牛顿方法,光滑方程组方法,可微无约束优化方法和内点方法等。 本文第3章对非线性互补问题的一阶微分方程方法进行了研究。将非线性互补问题转化为等价的优化问题,然后利用微分方程方法求解,基于平方函数空间变换构造了一个新型的解决非线性互补问题的微分方程系统。它的结构简单,易于计算,并且保证迭代点一但进入可行域,其轨迹便不再离开。在一定的条件下我们证明了非线性互补问题的解是该微分方程系统的平衡点,并且证明了该微分方程系统的稳定性和全局收敛性。最后我们给出了数值算例验证了该方法的有效性。 本文第4章对非线性互补问题的二阶微分方程方法进行了研究。应用了牛顿的方法建立了微分方程系统,证明了非线性互补问题的解是所构造微分方程系统的渐近稳定平衡点,并给出了算法,证明了算法具有二阶收敛速度。更多还原

【Abstract】 This dissertation is devoted to the study on differential equation approaches to nonlinear complementarity problems, including theories and numerical implementations of both first order and second order differential equation approaches.The class of complementarity problems is a important branch of mathematical programming. It provides a framework for both linear programming and quartic programming. It has an important relation with stationary point theories, variational inequality, linear and nonlinear analysis, and some applied mathematical problems such as economic and equilibrium problems. There are many effective algorithms, including stationary point methods, homotopy methods, project methods, Newton methods, smooth differential equation methods, differential unconstrainizing methods and interior point methods.In Chapter 3 of this paper, we present a first order differential equation system with barrier projection method for solving nonlinear complementarity problems. The main ideas lie in that nonlinear complementarity problems can be converted to optimization problems, and the resulting problems are solved by differential equation approaches. It possesses a simple structure for implementation in hardware and preserves feasibility. We prove that the solution of a nonlinear complementarity problem is exactly the equilibrium point of differential equation system, and prove the asymptotical stability and global convergence. In addition, numerical examples are reported to verify the validity of the differential equation system.In Chapter 4, we focus on second order differential equation system for solving nonlinear complementarity problems. We establish a differential equation system via Newton method, and prove that the solution of nonlinear complementarity problems is exact the equilibrium point of differential equation system. Finally, we present a numerical algorithm and demonstrate its local quadratic convergence rate.更多还原