Abstrakt
The absolute value equations (AVE) problem is an algebraic problem of solving Ax+|x| = b. So far, most of the research has focused on methods for solving AVE, but we address the problem itself by analyzing properties of AVE and the corresponding solution set.
In particular, we investigate topological properties of the solution set, such as convexity, boundedness, or connect-edness, or whether it consists of finitely many solutions. Further, we address problems related to the 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 polynomi-ally 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 were raised during the investigation of the problem.