线性矩阵不等式半定互补问题的数值求解方法
Numerical Methods for the Semidefinite Complementarity Problem with Linear Matrix Inequalities

线性矩阵不等式半定互补问题是一类新颖的多项式优化问题，是半定规划与互补问题的交叉研究内容。该问题可以转化为一系列带有线性矩阵不等式约束的多项式优化子问题，进而可以采用标量型松弛方法或矩阵型松弛方法进行求解。更进一步，当线性矩阵不等式半定互补问题的实数解个数有限时，利用本文给出的算法可以计算出该问题的全部实数解。最后，我们进行相关数值实验，分别使用标量型松弛方法和矩阵型松弛方法求解该问题，并将这两种方法的结果进行对比。
The semidefinite complementarity problem with linear matrix inequalities is a novel polynomial optimization problem, which is the cross-study content of semidefinite programming and complementarity problems. The problem can be transformed into a series of polynomial optimization subproblems with linear matrix inequality constraints. Then the problem can be solved by the scalar-type relaxation method or the matrix-type relaxation method. Furthermore, when the number of real solutions of the semidefinite complementarity problem with linear matrix inequalities is limited, all real solutions of the problem can be calculated by using the algorithm given in this paper. Finally, we conduct related numerical experiments. We use the scalar-type relaxation method and the matrix-type relaxation method to solve the problem and compare the results of these two methods.