水平线性互补问题投影迭代法和模系矩阵分裂迭代法的等价性

ON THE EQUIVALENCE OF PROJECTED ITERATIVE METHODS AND MODULUS-BASED MATRIX SPLITTING ITERATIVE METHODS FOR HORIZONTAL LINEAR COMPLEMENTARITY PROBLEMS

水平线性互补问题(HLCP)是著名线性互补问题(LCP)的重要推广形式之一,投影迭代法和模系矩阵分裂迭代法是最近提出的求解HLCP两类非常有效的热点方法.本文研究表明,尽管这两类方法导出原理不同,但在一定条件下是等价的.特别地,当模系矩阵分裂迭代法中参数矩阵Ω取为特定的正对角矩阵时,投影Jacobi法、投影Gauss-Seidel法和投影SOR法分别等价于模系Jacobi迭代法、加速的模系Gauss-Seidel迭代法和加速的模系SOR迭代法.此外,对一般的正对角矩阵Ω,本文也研究了两类方法的等价性.最后,通过数值算例验证了本文的理论结果.

Horizontal linear complementarity problem(HLCP)is one of the important generalization of the famous linear complementarity problem(LCP).The projected iterative method and the modulus-based matrix splitting iterative method are two recent proposed very effective methods for solving the HLCP.The research in this paper shows that although the deriving principles of these two methods are different,they are equivalent under certain conditions.In particular,when the parameter matrixΩin the modulus-based matrix splitting iteration methods is taken as a specific positive diagonal matrix,the projected Jacobi method,the projected Gauss-Seidel method and the projected SOR method are equivalent to the modulus-based Jacobi iteration method,the accelerated modulus-based Gauss-Seidel iteration method and the accelerated modulus-based SOR iteration method,respectively.In addition,for the general positive diagonal matrixΩ,the equivalence of these two methods is also studied.Finally,a numerical example is presented to verify the obtained theoretical results.