【作者】 李晖；

【导师】 张学胜；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2019， 硕士

【摘要】 互补问题在数学规划中占据着重要地位,与优化问题、变分原理、不动点理论等分支密切相关。张量特征值互补问题是互补问题与特征值问题的推广,本文主要研究偶数阶的对称张量特征值互补问题(OTEiCP)及其扩展对称张量高次特征值互补问题(OTHEiCP)的相关内容。本文首先介绍了张量特征值问题(TEiP),并且给定了判断张量正定(半正定)的充要条件。广义张量特征值问题(GTEiP)是TEiP的一个扩展,由此我们给出了广义Rayleigh商函数的定义,并指出广义Rayleigh商函数的稳定点即是对称GTEiP的解。对称张量特征值互补问题可转化为在标准单纯形上求解Rayleigh商函数的稳定点问题。更进一步,指出了OTEiCP的三种等价形式:分别以Rayleigh商、对数函数和多项式作为价值函数的一般非线性规划(NLP)。针对上述三种非线性规划,我们分别给出了相应的命题和定理:满足λ>0的NLP的稳定点是OTEiCP的解;NLP的最优解对应的互补特征值是OTEiCP的最大λ-解。本文还指出当且仅当(?)x≥ 0,s.t.Axm>0时,OTEiCP有解,并证明了当A取一些特殊张量时,相应的OTEiCP解的存在性。随后利用乘子法对OTEiCP进行求解,并给出数值算例,从而证明了该算法的有效性及可靠性。在对称张量特征值互补问题的基础上,我们还对OTHEiCP进行了探索。由于对称张量高次特征值互补问题结构的复杂性,其解的存在性条件更为严格。本文选取了OTHEiCP的2种特殊情况进行研究:当m=2时,OTHEiCP退化为二次特征值互补问题(QEiCP);在A=O,B正定或B=O,C正定的情形下,OTHEiCP可等价转化为一次的对称张量特征值互补问题,同时给出了上述OTHEiCP解的存在性定理。更多还原

【Abstract】 Complementarity problem plays an important role in mathematical programming,and is closely related to optimization problem,variational principle and fixed point theory.The tensor eigenvalue complementarity problem is a generalization of the complementarity problem and the eigenvalue problem.In this paper,we mainly study the even order symmetric tensor eigen-value complementarity problem(OTEiCP)and the symmetric tensor higher-degree eigenvalue complementarity problem(OTHEiCP).This paper first introduces the tensor eigenvalue problem(TEiP),and give a necessary and sufficient condition to determine the positive definiteness(or positive semidefiniteness)of ten-sor.The generalized tensor eigenvalue problem(GTEiP)is an extension of TEiP.We give the definition of generalized Rayleigh quotient function and point out that the stationary points of generalized Rayleigh quotient function are the solutions of the symmetric generalized tensor eigenvalue problem.The symmetric tensor eigenvalue complementarity problem can be transformed into finding a stationary point of the Rayleigh quotient function on the simplex.Furthermore,the OTEiCP is shown to be equivalent to three nonlinear programmings(NLP).The merit functions of NLP are Rayleigh quotient,logarithm and polynomial functions,respectively.In view of the above three nonlinear programmings,we give the corresponding propositions and theorems.In fact,every stationary point x of NLP with A(x)>0 is a solution of the OTEiCP and the optimal solution x of NLP with A(x)>0 is the maximum A-solution of the OTEiCP.This paper also points out that OTEiCP is solvable if and only if satisfying Axm>0 for some x≥ 0,and proves the solu-tion’s existence of OTEiCP under A taking some special tensors.Then,we propose a multiplier method for OTEiCP.The numerical experiments also show that the validity and reliability of the algorithm.On the basis of the symmetric tensor eigenvalue complementarity problem,we also explore OTHEiCP.Due to the complexity of the symmetric tensor higher-degree eigenvalue complemen-tarity problem,the existence conditions of solutions are more stringent.In this paper,two special cases of OTHEiCP are studied:when m＝2,OTHEiCP is transformed into the quadratic eigen-value complementary problem(QEiCP).In the case of A＝O,B is a positive definite tensor or B＝O,C is a positive definite tensor,OTHEiCP can be transformed into the symmetric tensor eigenvalue complementarity problem.At the same time,the existence theorem of the above OTHEiCP solutions is given.更多还原