绝对值方程研究进展

Advance in the study of absolute value equations

线性规划、二次规划、双矩阵对策等问题都能转化为线性互补问题,而线性互补问题又可以归结为绝对值方程,因此研究绝对值方程具有重要的意义。绝对值方程是一个NP-hard问题,对绝对值方程的研究现状进行了分析,给出了绝对值方程的理论研究现状,总结了绝对值方程的若干求解算法。这些算法可以归结为三类:1)逐次线性化方法,2)半光滑牛顿法,3)光滑牛顿法。指出解的存在性、构造光滑函数、采用智能算法求解以及算法收敛性分析将成为绝对值方程的研究热点。

The significance of the absolute value equations （AVE） arises from the fact that linear programs, quadratic programs, bimatrix games and other problems can all be reduced to the linear com- plementarity problem that in turn is equivalent to the AVE. AVE is an NP-hard problem in its general form. The current research situation of AVE was analyzed. Results of AVE in theory were given, and the algorithms for AVE can be summarized into three categories： 1 ） successive linearization method; 2） semi-smooth Newton method; 3 ） Smoothing Newton method. Combining with the author studies, it is concluded that ＂existence of solution, constructing smoothing function, using intelligent algorithms and convergence analysis＂ will be the research focuses on AVE.