An overview of absolute value equations: from theory to solution methods and challenges

This paper provides a comprehensive examination of the absolute value equations $$Ax-|x|=b$$ , a deceptively simple formulation that has attracted significant research interest in recent years. This problem is NP-hard and nondifferentiable, and it is closely related to the standard linear complementarity problem. Offering a comprehensive review of existing literature, the paper explores results concerning the existence and nonexistence of solutions to absolute value equations, as well as numerical algorithms developed to solve this complex equation. Beyond traditional solution methods, the paper investigates strategies for computing solutions of minimal norm, techniques for correcting infeasible systems, and other relevant topics. By identifying key challenges and highlighting open research questions, this paper provides valuable insights and guidance for shaping future research in this evolving and multifaceted field.