论文标题
绝对值方程和相关矩阵类的解决方案集的属性集
Properties of the solution set of absolute value equations and the related matrix classes
论文作者
论文摘要
绝对值方程（AVE）问题是解决AX+| X | = B的代数问题。到目前为止，大多数研究都集中在解决Aves的方法上，但是我们通过分析AVE的属性和相应的解决方案集来解决问题本身。特别是，我们研究了溶液集的拓扑特性，例如凸度，有限性，连接性，或者它是否由有限的解决方案组成。此外，我们解决了与解决方案的非负相关的问题，例如可溶性或独特的可溶性。 AVE可以通过不同的优化问题来制定，在这方面，我们对AVE的解决方案与Optima，Karush-Kuhn-Tucker点以及这些优化问题的可行解决方案感兴趣。 我们表征与上述属性关联的矩阵类，并检查识别问题的计算复杂性；其中一些类是多项式可识别的，但其他一些类是NP-HARD。对于棘手的情况，我们提出了各种足够的条件。我们还发布了在调查问题期间引起的新的具有挑战性的问题。
The absolute value equations (AVE) problem is an algebraic problem of solving Ax+|x|=b. So far, most of the research focused on methods for solving AVEs, but we address the problem itself by analysing properties of AVE and the corresponding solution set. In particular, we investigate topological properties of the solution set, such as convexity, boundedness, connectedness, or whether it consists of finitely many solutions. Further, we address problems related to nonnegativity of solutions such as solvability or unique solvability. AVE can be formulated by means of different optimization problems, and in this regard we are interested in how the solutions of AVE are related with optima, Karush-Kuhn-Tucker points and feasible solutions of these optimization problems. We characterize the matrix classes associated with the above mentioned properties and inspect the computational complexity of the recognition problem; some of the classes are polynomially recognizable, but some others are proved to be NP-hard. For the intractable cases, we propose various sufficient conditions. We also post new challenging problems that raised during the investigation of the problem.
