【作者】 姜勇；

【导师】 周光明；

【作者基本信息】 湘潭大学 ， 计算数学， 2015， 硕士

【摘要】 多项式优化问题是一类重要的优化问题,它已被广泛应用于信号处理和系统控制理论等领域的数学建模。因此,研究这类问题的求解方法具有重要意义。近来,J.B. Lasserre提出了一种求解多项式优化问题全局最优值的Lasserre松弛方法。该方法已吸引了许多优化学者的关注。本文的主要内容有两个,其一是较全面地测试由Henrion和Lasserre开发的基于Lasserre松弛方法的软件包GloptiPoly求解二次规划的数值表现,其二是探讨子问题基于Lasserre松弛方法求解的信赖域算法的数值表现。第一章简要介绍了多项式优化问题以及Lasserre松弛方法。第二章分别考察了软件包GloptiPoly求解随机生成的无约束二次规划、带线性约束和二次约束的二次规划的数值表现。数值结果表明,该软件包能较好地求解大部分中小规模的二次规划,包括目标函数近奇异的二次规划。第三章首先针对无约束非线性规划和带线性约束的非线性规划问题,提出了子问题基于Lasserre松弛方法求解的信赖域算法,接着分析了算法的收敛性,最后通过数值实验验证了算法的有效性。论文最后对全文做了简单的总结和展望。更多还原

【Abstract】 Polynomial optimization problem (POP) is a class of important optimization problem. It has been applied in many areas, such as signal processing, system control theory, as a kind of mathematical model.Therefore, how to solve the POP is an interesting problem. Recently, J.B. Lasserre proposed a method which is focused on finding out the global optimal value of POP. The method, for short Lasserre relaxation method, has been attracted attentions of many optimization experts.There are two main contents in this paper, one of which is to apply software package GloptiPoly, that was based on Lasserre relaxation method and was developed by Henrion and Lasserre, to solve quadratic programming and to test its numerical performances, other of which is to construct a trust domain method whose subproblem is solved by Lasserre relaxation method and to test its numerical performances.In Chapter 1, we briefly describe polynomial optimization problem and Lasserre relaxation method.In Chapter 2, we apply Lasserre relaxation method to solve various quadratic programming prob-lems, including unconstrained quadratic programming, linear constrained quadratic programming and quadratic constrained quadratic programming with stochastic data. Numerical results illustrate that GloptiPoly can efficiently solve most of stochastic numerical examples with middle and small scales, including almost singular examples.In Chapter 3, firstly, trust domain algorithms aiming at unconstrained nonlinear programming and linear constrained nonlinear programming are put forward. Secondly, convergence properties of these trust domain algorithms are given. Finally, numerical experiments show that the algorithms are effective.At last, simple summaries and expectations of the paper are made.更多还原