Optimization Letters (2021) 15:2241–2254 https://doi.org/10.1007/s11590-020-01691-z ORIGINAL PAPER A new concave minimization algorithm for the absolute value equation solution Moslem Zamani 1,2 · Milan Hladík 3 Received: 27 April 2020 / Accepted: 15 December 2020 / Published online: 5 January 2021 © The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature 2021 Abstract In this paper, we study the absolute value equation (AVE) Ax − b =|x |. One effective approach to handle AVE is by using concave minimization methods. We propose a new method based on concave minimization methods. We establish its finite convergence under mild conditions. We also study some classes of AVEs which are polynomial time solvable. Keywords Absolute value equation · Concave minimization algorithms · Linear complementarity problem 1 Introduction We consider the absolute value equation problem of finding an x ∈ R n such that Ax − b =|x |, (AVE) where A ∈ R n×n , b ∈ R n and |·| denotes absolute value. In general, (AVE) is an NP-hard problem [16]. Since a general linear complementarity problem can be formulated as an absolute value equation, several methods, such as Newton-like methods [3,15,31] or concave optimization methods [20,21], have been proposed for solving (AVE). B Moslem Zamani zamani.moslem@tdt.edu.vn Milan Hladík hladik@kam.mff.cuni.cz 1 Parametric MultiObjective Optimization Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam 2 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 3 Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 11800 Prague, Czech Republic 123