线性互补问题在宽邻域下的局部二次收敛算法

A Quadratically Convergent Algorithm in a Wide Neighborhood for Linear Complementarity Problems

艾文宝(2004)的宽邻域算法弥补了内点法在理论和实践表现之间的差异.基于这个算法的优越性,将其推广到线性互补问题中.新算法在一次迭代中,采用两个方向的线性组合作为新方向,并以满步长到达下一个点.可以证明,该算法具有O(n^(1/2)L)的理论复杂度,这是迄今为止最好的复杂度结果.同时,在假设线性互补问题存在严格互补解的前提下,证明算法具有局部二次收敛性.最后,数值实验说明算法是有效的.

The algorithm, proposed by AI（2004） in his new wide neighborhood, can fill the gap between in theory and in practice for primal-dual interior-point methods. Based on the point, it is generalized to solve linear complementarity problems. In each iteration of the algorithm of AI, a linear combined direction is used, and a full step size along the direction is adequate for the next iterate point. It can be proved that the algorithm has an O（n（1/2） L） complexity bound, which is the best result so far. Furthermore,its quadratic convergence can be shown under the assumptation that a strictly complementary solution exists. Numerical tests show that it is effective.