【作者】 朱琳；

【导师】 张新珍；

【作者基本信息】 天津大学 ， 数学， 2022， 硕士

【摘要】 带有二阶锥互补约束的多项式优化问题是带有二阶锥互补约束的数学规划问题的一个子类,也是非线性规划中一类重要问题,在工程、经济等领域有广泛应用.半定松弛算法是求解多项式优化问题的有效算法,在一定条件下半定松弛算法具有很好的收敛性,比如有限收敛性.如果能将原问题转化为一般的多项式优化问题,那么原问题就可以应用半定松弛算法进行求解.因此本论文要讨论求解带有二阶锥互补约束的多项式优化问题的半定松弛方法.这篇论文主要讨论了如何用Lasserre型的半定松弛方法求解带有二阶锥互补约束的多项式优化问题.对于带有二阶锥互补约束的多项式优化问题,其互补约束的等价转化形式并不唯一,而算法的规模与效率取决于多项式的维数与次数,因此考虑如何将原问题转化为低阶的多项式优化问题是求解这类问题的重点.本论文首先考虑该问题的一般形式,将问题重构为相对低阶的多项式优化问题,并应用Lasserre型的半定松弛方法对重构问题进行求解.其次本论文考虑该问题的一种特殊情况,对于这种形式我们将问题重构为更低阶的多项式优化问题,同样应用Lasserre型的半定松弛方法进行求解.我们证明了半定松弛算法在一定条件下具有有限收敛性.数值实验表明这两种模型与算法的有效.更多还原

【Abstract】 Polynomial optimization problem with second-order cone complementarity constraints is a subclass of mathematical programming problem with second-order cone complementarity constraint,and it is also an important problem in nonlinear programming,which is widely used in engineering,economy and other fields.Semidefinite relaxation algorithm is an e?ective algorithm for solving polynomial optimization problems.Under certain conditions,semidefinite relaxation algorithm has good convergence,such as finite convergence.If the original problem can be transformed into a general polynomial optimization problem,it can be solved by semidefinite relaxation method.So this paper discusses semidefinite relaxation method for polynomial optimization problems with second-order cone complementary constraints.This paper mainly discuss how to solve the polynomial optimization problem with second-order cone complementarity constraints by Lasserre’s type of semidefinite relaxation method.For polynomial optimization problems with second-order cone complementarity constraints,the equivalent transformation form of complementarity constraints is not unique,and the scale and e ciency of the algorithm depend on the dimensionality and degree of the polynomial,so how to transform the original problem into a low-order polynomial optimization problem is the key to solve this kind of problems.This paper firstly consider the general form of the problem,and the problem is reconstructed into a relatively low-order polynomial optimization problem.Then Lasserre’s type of semidefinite relaxation method is applied to solve the reconstruction problem.Secondly,a special case of this problem is considered in this paper.For this form,the problem is reconstructed into a polynomial optimization problem of lower order,and the Lasserre’s type of semidefinite relaxation method is also used to solve it.We prove that the semidefinite relaxation method has finite convergence under certain conditions.Numerical experiments show the e?ectiveness of these two models and algorithms.更多还原