【作者】 董建新；

【导师】 王希云；

【作者基本信息】 太原科技大学 ， 应用数学， 2010， 硕士

【摘要】 非线性互补问题在工程和经济中有许多重要应用,已经产生了大量的求解方法,同时也取得了全局收敛性和局部超线性收敛性结果.然而,学者们多采用NCP函数把非线性互补问题转化为非光滑方程组来求解.本文采用Kanzow光滑逼近函数逼近Fischer-Burmeister函数,从而得到相应的光滑非线性方程组,并将其转化为优化问题,再通过引入自适应技术和非单调技术,与原来的较为可靠、稳定的信赖域方法相结合,提出了两种新的求解方法.在第三章中,将自适应技术与信赖域算法结合提出了求解非线性互补问题的自适应光滑信赖域算法.该算法使得信赖域半径可以根据当前迭代点的参数进行自我调整;同时,若目标函数下降量足够大,则更新光滑逼近函数的光滑化系数.第四章在第三章算法的基础上引入非单调技术,从而给出求解非线性互补问题的非单调自适应光滑信赖域算法.该算法在信赖域子问题的下降量估计中引入了“非单调比率”,当该比率可接受时接受该步迭代;同时,若目标函数下降量足够大,则更新光滑逼近函数的光滑化系数.在假设F是P0函数的条件下,文中还证明了两种算法所产生的点列包含在一个水平集中;且在水平集为紧集的前提下,算法至少产生一个聚点,从而证明了算法的全局收敛性.进一步地,本文还给出了算法产生的点列在一定条件下有超线性收敛性及二次收敛性等性质.在两种算法理论证明的基础上,第五章进行了数值实验,给出了比较结果,得到算法的有效性.更多还原

【Abstract】 The nonlinear complementarity problem (NCP) has many important applications in engineering and economics, and there have developed many numerical methods to solve it at the same time, global and local convergence results have also been obtained. In the last years more attention has been devoted to reformulating the nonlinear complementarity problem as a system of nonsmooth equations by using some NCP function.In this paper, we use the Kanzow function to approximate the Fischer-Burmeister function so that smooth and nonlinear equatons can be obtained and then changed into a optimization problem. With the combination of the adaptive technique、the nonmonotone technique and the trust-region method which is more stable and reliable than linear search method, two new algorithms for solving these equations are proposed in Chapter 3 and Chapter 4.In Chapter 3, we combine the trust region subproblem with the adptive techique to propose a smoothing adptive trust region algorithm for the nonlinear complementarity problem. With the adaptive technique, the radius of trust regionΔk can be adjusted automatically to improve the efficiency of trust region methods. On the other hand, the smoothing coefficient of the Kanzow function is updated when the object function gets some satisfying reduction.In Chapter 4, based on the algorithm in Chapter 3, the new algorithm adopts a ’nonmontone ratio’ to approximate the reduction of the object functions. An iteration is accepted when the ratio is not very small. On the other hand, the smoothing coefficient of the Kanzow function is updated when the object function gets some satisfying reduction.With the assumption that F is a P0 function, we prove that the sequence generated by the algorithm remains in some level set. Furthermore, the method will generate at least one accumulation point if the level set is compact, which leads to the global convergence. In addition, the sequence will converge to one point and the superlinear convergence or even quadratic convergence under some conditions.In Chapter 5, a series of numerical examples are tested, which shows the algorithm is quite promising.更多还原