Modified Exact Jacobian Semidefinite Programming Relaxation for Celis-Dennis-Tapia Problem

A modified exact Jacobian semidefinite programming(SDP)relaxation method is proposed in this paper to solve the Celis-Dennis-Tapia(CDT)problem using the Jacobian matrix of objective and constraining polynomials.In the modified relaxation problem,the number of introduced constraints and the lowest relaxation order decreases significantly.At the same time,the finite convergence property is guaranteed.In addition,the proposed method can be applied to the quadratically constrained problem with two quadratic constraints.Moreover,the efficiency of the proposed method is verified by numerical experiments.

线性矩阵不等式半定互补问题的数值求解方法

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。