锥规划的光滑算法研究

发布时间：2018-07-01 09:08

本文选题：二阶锥规划 + 线性规划　；参考：《内蒙古大学》2017年硕士论文

【摘要】：线性规划问题是研究变量在仿射集和凸多面体交集上的一类凸优化问题.作为线性规划的推广,二阶锥规划也是一类凸优化问题,它是在一个仿射子空间和有限个二阶锥的笛卡尔乘积的交集上极大化或极小化一个线性函数.许多数学规划问题,都可以转化为二阶锥问题求解.线性规划和二阶锥规划在工程、控制与设计等诸多领域的广泛应用,使其成为数学规划的一个重要研究方向.本文主要研究线性规划和二阶锥规划的光滑牛顿法.全文共分为四章.第一章,介绍线性规划和二阶锥规划的研究背景及现状.第二章,通过光滑逼近Fischer-Burmeister函数,构造出一个新的光滑函数,得出该函数的连续可微性.基此给出一个求解线性规划问题的光滑牛顿法.此外,证明了算法的全局收敛性.在解点处雅可比矩阵可逆的条件下,得到算法的二次收敛速度.最后通过数值实验证明了算法的有效性.第三章,通过对称扰动Fischer-Burmeister函数,提出一个新的互补函数.基于该函数,把二阶锥规划问题转化为一个参数化的光滑方程组,并利用光滑牛顿法求解.此外,证明了算法的全局收敛性.在解点处雅可比矩阵可逆的条件下,得到算法的二次收敛速度.最后进行数值实验,数值结果表明了算法的有效性.第四章是对本文的总结.
[Abstract]:Linear programming problem is a class of convex optimization problems for variables on affine sets and convex polyhedron intersection. As a generalization of linear programming, second-order cone programming is also a class of convex optimization problems. It is a linear function that is maximized or minimized on the intersection of the Cartesian product of an affine subspace and the finite second-order cone. Many mathematical programming problems can be transformed into second-order cone problems. Linear programming and second-order cone programming are widely used in many fields such as engineering, control and design, which make them become an important research direction of mathematical programming. In this paper, the smooth Newton method for linear programming and second order cone programming is studied. The full text is divided into four chapters. The first chapter introduces the research background and present situation of linear programming and second-order cone programming. In chapter 2, a new smooth function is constructed by smoothing the Fischer-Burmeister function, and the continuous differentiability of the function is obtained. Based on this, a smooth Newton method for solving linear programming problems is given. In addition, the global convergence of the algorithm is proved. The quadratic convergence rate of the algorithm is obtained under the condition that Jacobian matrix is reversible at the solution point. Finally, the effectiveness of the algorithm is proved by numerical experiments. In chapter 3, a new complementary function is proposed by symmetric perturbation Fischer-Burmeister function. Based on this function, the second order cone programming problem is transformed into a parameterized smooth equation system and solved by the smooth Newton method. In addition, the global convergence of the algorithm is proved. The quadratic convergence rate of the algorithm is obtained under the condition that Jacobian matrix is reversible at the solution point. Finally, numerical experiments are carried out, and the numerical results show the effectiveness of the algorithm. The fourth chapter is the summary of this paper.
【学位授予单位】：内蒙古大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O221

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