圆锥规划和圆锥互补问题的光滑牛顿法研究

发布时间：2018-03-11 15:40

本文选题：圆锥规划　切入点：圆锥互补问题　出处：《桂林电子科技大学》2017年硕士论文　论文类型：学位论文

【摘要】：圆锥规划和圆锥互补问题是数学规划领域的一个重要分支.圆锥规划是在有限个圆锥笛卡尔积和仿射子空间的交集上求目标函数的极小值或极大值问题,而圆锥互补问题是一类均衡优化问题.圆锥规划和圆锥互补问题被广泛应用于工程问题,如多指手臂机器人的最优抓力操纵问题、接触力优化问题和四足机器人力优化问题.但在标准内积下,圆锥通常是非对称锥,因此目前关于圆锥规划和圆锥互补问题的算法尚不多见.本文主要给出圆锥规划和圆锥互补问题的光滑牛顿法,并取得以下主要成果:1.基于一个光滑函数和圆锥与二阶锥的代数关系,给出求解圆锥规划的单调光滑牛顿法.运用欧几里得约当代数理论,分析了算法的全局和局部二阶收敛性.四足机器人的力优化问题和随机生成的圆锥规划问题的数值结果表明新算法的有效性.2.给出求解圆锥规划的非单调光滑牛顿法.为了提高新算法的收敛速度,在光滑牛顿法中引入非单调线搜索.在适当的假设下,证明了新算法是全局和局部二阶收敛的.数值结果表明算法求解圆锥规划问题所需的计算时间和迭代次数都很少,且比较稳定,从而说明其有效性.3.基于一类带参数的光滑函数,将圆锥互补问题转化成非线性方程组,给出求解圆锥互补问题的非单调光滑牛顿法.该算法中引入新的非单调线搜索,以取得更好的数值结果.在适定的条件下证明效益函数的强制性以及算法的全局和局部二阶收敛性.通过数值结果验证算法的有效性.
[Abstract]:Cone programming and cone complementarity problems are important branches in the field of mathematical programming. Cone programming is a problem of finding the minimum or maximum value of objective functions on the intersection of finite cone Cartesian products and affine subspaces. Conical complementarity problem is a class of equilibrium optimization problems. Cone programming and cone complementarity problems are widely used in engineering problems, such as the optimal manipulating of multi-fingered manipulators. The contact force optimization problem and the quadruped robot force optimization problem. But under the standard inner product, the cone is usually asymmetric cone, Therefore, there are few algorithms for cone programming and cone complementarity problems. In this paper, the smooth Newton method for cone programming and cone complementarity problems is presented. On the basis of a smooth function and the algebraic relation between a cone and a second order cone, a monotone smooth Newton method for solving cone programming is given. The Euclidean approximate contemporary number theory is used. The global and local second order convergence of the algorithm is analyzed. The numerical results of the force optimization problem for quadruped robot and the randomly generated cone programming problem show the effectiveness of the new algorithm. 2. The nonmonotone smooth cattle for solving the cone programming is given. In order to improve the convergence rate of the new algorithm, Non-monotone line search is introduced into the smooth Newton method. Under appropriate assumptions, it is proved that the new algorithm is global and locally second-order convergent. The numerical results show that the computational time and the number of iterations required for the algorithm to solve the conical programming problem are few. Based on a class of smooth functions with parameters, the cone complementarity problem is transformed into nonlinear equations. A nonmonotone smooth Newton method for solving conical complementarity problems is presented, in which a new nonmonotone line search is introduced. In order to obtain better numerical results, the mandatory benefit function and the global and local second-order convergence of the algorithm are proved under suitable conditions. The validity of the algorithm is verified by numerical results.
【学位授予单位】：桂林电子科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O221

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