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The absolute value equations (AVE) problem is an algebraic problem of solvingAx+|x|=b. So far, most of the research focused on methods for solving AVEs, butwe address the problem itself by analysing properties of AVE and thecorresponding solution set. In particular, we investigate topologicalproperties of the solution set, such as convexity, boundedness, connectedness,or whether it consists of finitely many solutions. Further, we address problemsrelated to nonnegativity of solutions such as solvability or uniquesolvability. AVE can be formulated by means of different optimization problems,and in this regard we are interested in how the solutions of AVE are relatedwith optima, Karush-Kuhn-Tucker points and feasible solutions of theseoptimization problems.  We characterize the matrix classes associated with the above mentionedproperties and inspect the computational complexity of the recognition problem;some of the classes are polynomially recognizable, but some others are provedto be NP-hard. For the intractable cases, we propose various sufficientconditions. We also post new challenging problems that raised during theinvestigation of the problem.

Properties of the solution set of absolute value equations and the related matrix classes