基于凝聚函数的和声搜索算法求解绝对值方程*

绝对值方程Ax-|x|=b是一个不可微的NP-hard问题。在假设矩阵A的奇异值大于1（这里矩阵A的奇异值定义为矩阵ATA特征值的非负平方根）时,给出了求解绝对值方程一个新的光滑化算法。通过引入一种凝聚函数对绝对值方程进行光滑化处理,得到一个非线性方程组;再引入适当的目标函数,进而把绝对值方程化为无约束优化问题,然后利用和声搜索算法对其进行求解。该算法模拟了音乐创作中乐师们凭借自己的记忆,通过反复调整乐队中各乐器的音调,最终达到一个美妙的和声状态的过程。数值结果表明,该算法收敛快,数值稳定性好,是求解绝对

Harmony search algorithm for absolute value equations based on aggregate function

Absolute value equations Ax-|x|=b is a non-differentiable NP-hard problem in its general form. This paper proposed a new smoothing method for solving absolute value equations under the condition that all singular values of A exceed one. First, using the aggregate function, transformed the absolute value equations problem into smooth nonlinear equations, furthermore, an unconstrained differentiable optimization problem by introducing appropriate objective function. Then applied harmony search algorithm to solve this problem. This algorithm simulated the music of the musicians with their own memory, by repeatedly adjusting the pitch of the instruments in the band; eventually reach a wonderful state of the process of sound. The numerical results show that the method is feasible and effective to absolute value equations.