{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T16:33:13Z","timestamp":1775579593458,"version":"3.50.1"},"reference-count":149,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,8,8]],"date-time":"2025-08-08T00:00:00Z","timestamp":1754611200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,8,8]],"date-time":"2025-08-08T00:00:00Z","timestamp":1754611200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100007397","name":"Charles University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100007397","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    This paper provides a comprehensive examination of the absolute value equations\n                    <jats:inline-formula>\n                      <jats:tex-math>$$Ax-|x|=b$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , 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.\n                  <\/jats:p>","DOI":"10.1007\/s10589-025-00717-5","type":"journal-article","created":{"date-parts":[[2025,8,8]],"date-time":"2025-08-08T08:41:42Z","timestamp":1754642502000},"page":"435-488","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["An overview of absolute value equations: from theory to solution methods and challenges"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7340-8491","authenticated-orcid":false,"given":"Milan","family":"Hlad\u00edk","sequence":"first","affiliation":[]},{"given":"Hossein","family":"Moosaei","sequence":"additional","affiliation":[]},{"given":"Fakhrodin","family":"Hashemi","sequence":"additional","affiliation":[]},{"given":"Saeed","family":"Ketabchi","sequence":"additional","affiliation":[]},{"given":"Panos M.","family":"Pardalos","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,8,8]]},"reference":[{"issue":"1","key":"717_CR1","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1137\/070681867","volume":"30","author":"L Brugnano","year":"2008","unstructured":"Brugnano, L., Casulli, V.: Iterative solution of piecewise linear systems. SIAM J. Sci. Comput. 30(1), 463\u2013472 (2008). https:\/\/doi.org\/10.1137\/070681867","journal-title":"SIAM J. Sci. Comput."},{"issue":"2\u20133","key":"717_CR2","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/j.laa.2006.05.004","volume":"419","author":"OL Mangasarian","year":"2006","unstructured":"Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419(2\u20133), 359\u2013367 (2006)","journal-title":"Linear Algebra Appl."},{"key":"717_CR3","unstructured":"Rohn, J.: Proofs to \u201cSolving interval linear systems\u201d. Freiburger Intervall-Berichte 84\/7, Albert-Ludwigs-Universit\u00e4t, Freiburg (1984)"},{"key":"717_CR4","unstructured":"Rohn, J.: Solving interval linear systems. Freiburger Intervall-Berichte 84\/7, Albert-Ludwigs-Universit\u00e4t, Freiburg (1984)"},{"issue":"C","key":"717_CR5","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/0024-3795(89)90004-9","volume":"126","author":"J Rohn","year":"1989","unstructured":"Rohn, J.: Systems of linear interval equations. Linear Algebra Appl. 126(C), 39\u201378 (1989)","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"717_CR6","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/s10589-006-0395-5","volume":"36","author":"OL Mangasarian","year":"2007","unstructured":"Mangasarian, O.L.: Absolute value programming. Comput. Optim. Appl. 36(1), 43\u201353 (2007)","journal-title":"Comput. Optim. Appl."},{"key":"717_CR7","doi-asserted-by":"publisher","first-page":"102906","DOI":"10.1016\/j.scico.2022.102906","volume":"225","author":"L Chen","year":"2023","unstructured":"Chen, L., Wei, D., Yin, B., Wang, J.: Static analysis of linear absolute value equalities among variables of a program. Sci. Comput. Program. 225, 102906 (2023). https:\/\/doi.org\/10.1016\/j.scico.2022.102906","journal-title":"Sci. Comput. Program."},{"key":"717_CR8","doi-asserted-by":"crossref","unstructured":"Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem, Revised ed. of the 1992 original edn. SIAM, Philadelphia, PA (2009)","DOI":"10.1137\/1.9780898719000"},{"key":"717_CR9","first-page":"1","volume-title":"Encyclopedia of Optimization","author":"PM Pardalos","year":"2024","unstructured":"Pardalos, P.M.: LCP: Pardalos-Rosen mixed integer formulation. In: Pardalos, P.M., Prokopyev, O.A. (eds.) Encyclopedia of Optimization, 3rd edn., pp. 1\u20132. Springer, Cham (2024)","edition":"3"},{"key":"717_CR10","doi-asserted-by":"publisher","first-page":"500","DOI":"10.13001\/1081-3810.1327","volume":"18","author":"J Rohn","year":"2009","unstructured":"Rohn, J.: Forty necessary and sufficient conditions for regularity of interval matrices: A survey. Electron. J. Linear Algebra 18, 500\u2013512 (2009)","journal-title":"Electron. J. Linear Algebra"},{"key":"717_CR11","doi-asserted-by":"publisher","first-page":"500","DOI":"10.1016\/j.laa.2014.12.017","volume":"471","author":"A Griewank","year":"2015","unstructured":"Griewank, A., Bernt, J.-U., Radons, M., Streubel, T.: Solving piecewise linear systems in abs-normal form. Linear Algebra Appl. 471, 500\u2013530 (2015). https:\/\/doi.org\/10.1016\/j.laa.2014.12.017","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"717_CR12","first-page":"297","volume":"43","author":"S Ovchinnikov","year":"2002","unstructured":"Ovchinnikov, S.: Max-min representation of piecewise linear functions. Beitr. Algebra Geom. 43(1), 297\u2013302 (2002)","journal-title":"Beitr. Algebra Geom."},{"key":"717_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-4340-7","volume-title":"Introduction to Piecewise Differentiable Equations. SpringerBriefs Optim","author":"S Scholtes","year":"2012","unstructured":"Scholtes, S.: Introduction to Piecewise Differentiable Equations. SpringerBriefs Optim. Springer, Berlin (2012)"},{"issue":"1","key":"717_CR14","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1109\/31.1705","volume":"35","author":"LO Chua","year":"1988","unstructured":"Chua, L.O., Deng, A.-C.: Canonical piecewise-linear representation. IEEE Trans. Circuits Syst. 35(1), 101\u2013111 (1988). https:\/\/doi.org\/10.1109\/31.1705","journal-title":"IEEE Trans. Circuits Syst."},{"issue":"1","key":"717_CR15","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1109\/72.363451","volume":"6","author":"J-N Lin","year":"1995","unstructured":"Lin, J.-N., Unbehauen, R.: Canonical piecewise-linear networks. IEEE Trans. Neural Netw. 6(1), 43\u201350 (1995). https:\/\/doi.org\/10.1109\/72.363451","journal-title":"IEEE Trans. Neural Netw."},{"issue":"3","key":"717_CR16","doi-asserted-by":"publisher","first-page":"1858","DOI":"10.1137\/08072749X","volume":"31","author":"L Brugnano","year":"2009","unstructured":"Brugnano, L., Casulli, V.: Iterative solution of piecewise linear systems and applications to flows in porous media. SIAM J. Sci. Comput. 31(3), 1858\u20131873 (2009). https:\/\/doi.org\/10.1137\/08072749X","journal-title":"SIAM J. Sci. Comput."},{"issue":"16","key":"717_CR17","doi-asserted-by":"publisher","first-page":"3937","DOI":"10.1016\/j.cam.2012.02.042","volume":"236","author":"V Casulli","year":"2012","unstructured":"Casulli, V., Zanolli, P.: Iterative solutions of mildly nonlinear systems. J. Comput. Appl. Math. 236(16), 3937\u20133947 (2012). https:\/\/doi.org\/10.1016\/j.cam.2012.02.042","journal-title":"J. Comput. Appl. Math."},{"issue":"4","key":"717_CR18","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1080\/09720502.2014.996022","volume":"18","author":"L Yong","year":"2015","unstructured":"Yong, L.: Iteration method for absolute value equation and applications in two-point boundary value problem of linear differential equation. J. Interdiscip. Math. 18(4), 355\u2013374 (2015)","journal-title":"J. Interdiscip. Math."},{"issue":"4","key":"717_CR19","first-page":"63","volume":"80","author":"MA Noor","year":"2018","unstructured":"Noor, M.A., Noor, K.I., Batool, S.: On generalized absolute value equations. UPB Sci. Bull. Ser. A 80(4), 63\u201370 (2018)","journal-title":"UPB Sci. Bull. Ser. A"},{"issue":"1","key":"717_CR20","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1137\/22M1497018","volume":"44","author":"M Hlad\u00edk","year":"2023","unstructured":"Hlad\u00edk, M.: Properties of the solution set of absolute value equations and the related matrix classes. SIAM J. Matrix Anal. Appl. 44(1), 175\u2013195 (2023). https:\/\/doi.org\/10.1137\/22M1497018","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"717_CR21","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1007\/s10589-007-9158-1","volume":"44","author":"O Prokopyev","year":"2009","unstructured":"Prokopyev, O.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44(3), 363 (2009)","journal-title":"Comput. Optim. Appl."},{"issue":"7","key":"717_CR22","doi-asserted-by":"publisher","first-page":"1469","DOI":"10.1007\/s11590-015-0893-4","volume":"9","author":"OL Mangasarian","year":"2015","unstructured":"Mangasarian, O.L.: A hybrid algorithm for solving the absolute value equation. Optim. Lett. 9(7), 1469\u20131474 (2015)","journal-title":"Optim. Lett."},{"issue":"6","key":"717_CR23","doi-asserted-by":"publisher","first-page":"2241","DOI":"10.1007\/s11590-020-01691-z","volume":"15","author":"M Zamani","year":"2021","unstructured":"Zamani, M., Hlad\u00edk, M.: A new concave minimization algorithm for the absolute value equation solution. Optim. Lett. 15(6), 2241\u20132254 (2021)","journal-title":"Optim. Lett."},{"issue":"1","key":"717_CR24","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1007\/s10589-017-9939-0","volume":"69","author":"M Hlad\u00edk","year":"2018","unstructured":"Hlad\u00edk, M.: Bounds for the solutions of absolute value equations. Comput. Optim. Appl. 69(1), 243\u2013266 (2018)","journal-title":"Comput. Optim. Appl."},{"key":"717_CR25","doi-asserted-by":"publisher","unstructured":"Kumar, S., Deepmala, Hlad\u00edk, M., Moosaei, H.: Characterization of unique solvability of absolute value equations: An overview, extensions, and future directions (2024). https:\/\/doi.org\/10.1007\/s11590-024-02094-0","DOI":"10.1007\/s11590-024-02094-0"},{"key":"717_CR26","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.aml.2017.08.012","volume":"76","author":"S-L Wu","year":"2018","unstructured":"Wu, S.-L., Li, C.-X.: The unique solution of the absolute value equations. Appl. Math. Lett. 76, 195\u2013200 (2018)","journal-title":"Appl. Math. Lett."},{"key":"717_CR27","doi-asserted-by":"publisher","first-page":"2017","DOI":"10.1007\/s11590-020-01672-2","volume":"15","author":"S Wu","year":"2021","unstructured":"Wu, S., Shen, S.: On the unique solution of the generalized absolute value equation. Optim. Lett. 15, 2017\u20132024 (2021)","journal-title":"Optim. Lett."},{"key":"717_CR28","doi-asserted-by":"publisher","first-page":"190","DOI":"10.13001\/1081-3810.1086","volume":"9","author":"SM Fallat","year":"2002","unstructured":"Fallat, S.M., Tsatsomeros, M.J.: On the Cayley transform of positivity classes of matrices. Electron. J. Linear Algebra 9, 190\u2013196 (2002)","journal-title":"Electron. J. Linear Algebra"},{"issue":"1","key":"717_CR29","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s11590-006-0005-6","volume":"1","author":"OL Mangasarian","year":"2007","unstructured":"Mangasarian, O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1(1), 3\u20138 (2007)","journal-title":"Optim. Lett."},{"issue":"7","key":"717_CR30","doi-asserted-by":"publisher","first-page":"1469","DOI":"10.1007\/s11590-017-1115-z","volume":"11","author":"OL Mangasarian","year":"2017","unstructured":"Mangasarian, O.L.: Sufficient conditions for the unsolvability and solvability of the absolute value equation. Optim. Lett. 11(7), 1469\u20131475 (2017)","journal-title":"Optim. Lett."},{"issue":"4","key":"717_CR31","doi-asserted-by":"publisher","first-page":"1645","DOI":"10.1137\/22M1517184","volume":"44","author":"M Radons","year":"2023","unstructured":"Radons, M., Tonelli-Cueto, J.: Generalized Perron roots and solvability of the absolute value equation. SIAM J. Matrix Anal. Appl. 44(4), 1645\u20131666 (2023). https:\/\/doi.org\/10.1137\/22M1517184","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"5","key":"717_CR32","doi-asserted-by":"publisher","first-page":"1017","DOI":"10.1007\/s11590-011-0318-y","volume":"6","author":"J Rohn","year":"2012","unstructured":"Rohn, J.: On Rump\u2019s characterization of P-matrices. Optim. Lett. 6(5), 1017\u20131020 (2012)","journal-title":"Optim. Lett."},{"key":"717_CR33","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1016\/S0024-3795(01)00590-0","volume":"363","author":"SM Rump","year":"2003","unstructured":"Rump, S.M.: On P-matrices. Linear Algebra Appl. 363, 237\u2013250 (2003)","journal-title":"Linear Algebra Appl."},{"issue":"6","key":"717_CR34","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1080\/0308108042000220686","volume":"52","author":"J Rohn","year":"2004","unstructured":"Rohn, J.: A theorem of the alternatives for the equation $$Ax+ B| x|= b$$. Linear Multilinear Algebra 52(6), 421\u2013426 (2004)","journal-title":"Linear Multilinear Algebra"},{"issue":"2","key":"717_CR35","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/s10957-009-9557-9","volume":"143","author":"C Zhang","year":"2009","unstructured":"Zhang, C., Wei, Q.J.: Global and finite convergence of a generalized Newton method for absolute value equations. J. Optim. Theory Appl. 143(2), 391\u2013403 (2009)","journal-title":"J. Optim. Theory Appl."},{"issue":"1","key":"717_CR36","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1007\/BF01582570","volume":"64","author":"GE Coxson","year":"1994","unstructured":"Coxson, G.E.: The P-matrix problem is co-NP-complete. Math. Program. 64(1), 173\u2013178 (1994)","journal-title":"Math. Program."},{"issue":"1","key":"717_CR37","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/s11590-012-0560-y","volume":"8","author":"J Rohn","year":"2014","unstructured":"Rohn, J., Hooshyarbakhsh, V., Farhadsefat, R.: An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optim. Lett. 8(1), 35\u201344 (2014)","journal-title":"Optim. Lett."},{"issue":"2","key":"717_CR38","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s10107-010-0439-6","volume":"134","author":"IB Gharbia","year":"2012","unstructured":"Gharbia, I.B., Gilbert, J.C.: Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix. Math. Program. 134(2), 349\u2013364 (2012)","journal-title":"Math. Program."},{"issue":"2","key":"717_CR39","doi-asserted-by":"publisher","first-page":"705","DOI":"10.1007\/s10957-015-0845-2","volume":"169","author":"S-L Wu","year":"2016","unstructured":"Wu, S.-L., Guo, P.: On the unique solvability of the absolute value equation. J. Optim. Theory Appl. 169(2), 705\u2013712 (2016)","journal-title":"J. Optim. Theory Appl."},{"issue":"114","key":"717_CR40","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1090\/S0025-5718-1971-0301955-6","volume":"25","author":"JR Kuttler","year":"1971","unstructured":"Kuttler, J.R.: A fourth-order finite-difference approximation for the fixed membrane eigenproblem. Math. Comput. 25(114), 237\u2013256 (1971)","journal-title":"Math. Comput."},{"issue":"5","key":"717_CR41","doi-asserted-by":"publisher","first-page":"953","DOI":"10.1016\/j.laa.2011.02.016","volume":"435","author":"CR Johnson","year":"2011","unstructured":"Johnson, C.R., Smith, R.L.: Inverse M-matrices, II. Linear Algebra Appl. 435(5), 953\u2013983 (2011)","journal-title":"Linear Algebra Appl."},{"key":"717_CR42","unstructured":"Rohn, J.: A class of explicitly solvable absolute value equations. Technical report V-1202, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2014). https:\/\/asep.lib.cas.cz\/arl-cav\/cs\/detail-cav_un_epca-0424937-A-Class-of-Explicitly-Solvable-Absolute-Value-Equations\/"},{"key":"717_CR43","unstructured":"Rohn, J.: Overdetermined absolute value equations. Technical report V-1265, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2019). https:\/\/asep.lib.cas.cz\/arl-cav\/cs\/detail-cav_un_epca-0503902-Overdetermined-Absolute-Value-Equations\/"},{"key":"717_CR44","doi-asserted-by":"publisher","first-page":"107029","DOI":"10.1016\/j.aml.2021.107029","volume":"116","author":"S-L Wu","year":"2021","unstructured":"Wu, S.-L.: The unique solution of a class of the new generalized absolute value equation. Appl. Math. Lett. 116, 107029 (2021)","journal-title":"Appl. Math. Lett."},{"key":"717_CR45","doi-asserted-by":"publisher","first-page":"106818","DOI":"10.1016\/j.aml.2020.106818","volume":"112","author":"B Hashemi","year":"2021","unstructured":"Hashemi, B.: Sufficient conditions for the solvability of a Sylvester-like absolute value matrix equation. Appl. Math. Lett. 112, 106818 (2021)","journal-title":"Appl. Math. Lett."},{"key":"717_CR46","doi-asserted-by":"publisher","first-page":"106966","DOI":"10.1016\/j.aml.2020.106966","volume":"116","author":"L-M Wang","year":"2021","unstructured":"Wang, L.-M., Li, C.-X.: New sufficient conditions for the unique solution of a square Sylvester-like absolute value equation. Appl. Math. Lett. 116, 106966 (2021)","journal-title":"Appl. Math. Lett."},{"issue":"1","key":"717_CR47","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/s10957-021-01972-2","volume":"192","author":"L-B Cui","year":"2022","unstructured":"Cui, L.-B., Fan, Y.-D., Song, Y.-S., Wu, S.-L.: The existence and uniqueness of solution for tensor complementarity problem and related systems. J. Optim. Theory Appl. 192(1), 321\u2013334 (2022)","journal-title":"J. Optim. Theory Appl."},{"key":"717_CR48","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/s10898-013-0076-8","volume":"58","author":"P Gajardo","year":"2014","unstructured":"Gajardo, P., Seeger, A.: Equilibrium problems involving the Lorentz cone. J. Global Optim. 58, 321\u2013340 (2014)","journal-title":"J. Global Optim."},{"issue":"5","key":"717_CR49","doi-asserted-by":"publisher","first-page":"1490","DOI":"10.1016\/j.cam.2010.08.036","volume":"235","author":"S-L Hu","year":"2011","unstructured":"Hu, S.-L., Huang, Z.-H., Zhang, Q.: A generalized Newton method for absolute value equations associated with second order cones. J. Comput. Appl. Math. 235(5), 1490\u20131501 (2011)","journal-title":"J. Comput. Appl. Math."},{"key":"717_CR50","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1016\/j.apnum.2021.07.020","volume":"170","author":"Z Jiang","year":"2021","unstructured":"Jiang, Z., Li, J.: Solving tensor absolute value equation. Appl. Numer. Math. 170, 255\u2013268 (2021). https:\/\/doi.org\/10.1016\/j.apnum.2021.07.020","journal-title":"Appl. Numer. Math."},{"key":"717_CR51","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1016\/j.amc.2015.07.064","volume":"269","author":"X-H Miao","year":"2015","unstructured":"Miao, X.-H., Yang, J., Hu, S.: A generalized Newton method for absolute value equations associated with circular cones. Appl. Math. Comput. 269, 155\u2013168 (2015)","journal-title":"Appl. Math. Comput."},{"key":"717_CR52","doi-asserted-by":"publisher","first-page":"106462","DOI":"10.1016\/j.aml.2020.106462","volume":"107","author":"F Mezzadri","year":"2020","unstructured":"Mezzadri, F.: On the solution of general absolute value equations. Appl. Math. Lett. 107, 106462 (2020)","journal-title":"Appl. Math. Lett."},{"key":"717_CR53","first-page":"1","volume":"1","author":"S-L Wu","year":"2019","unstructured":"Wu, S.-L., Li, C.-X.: A note on unique solvability of the absolute value equation. Optim. Lett. 1, 1\u20134 (2019)","journal-title":"Optim. Lett."},{"issue":"2","key":"717_CR54","first-page":"112","volume":"48","author":"A Mohamed","year":"2019","unstructured":"Mohamed, A.: On the unique solvability and numerical study of absolute value equations. Revue d\u2019analyse num\u00e9rique et de la th\u00e9orie de l\u2019approximation 48(2), 112\u2013121 (2019)","journal-title":"Revue d\u2019analyse num\u00e9rique et de la th\u00e9orie de l\u2019approximation"},{"key":"717_CR55","unstructured":"Rohn, J.: A reduction theorem for absolute value equations. Technical report V-1204, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2014). https:\/\/asep.lib.cas.cz\/arl-cav\/cs\/detail-cav_un_epca-0425071-A-Reduction-Theorem-for-Absolute-Value-Equations\/"},{"issue":"3","key":"717_CR56","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1007\/s11590-011-0284-4","volume":"6","author":"J Rohn","year":"2012","unstructured":"Rohn, J.: A theorem of the alternatives for the equation$$|Ax| -|B x| = b$$. Optim. Lett. 6(3), 585\u2013591 (2012)","journal-title":"Optim. Lett."},{"issue":"3","key":"717_CR57","doi-asserted-by":"publisher","first-page":"425","DOI":"10.2298\/YJOR220515036K","volume":"33","author":"S Kumar","year":"2023","unstructured":"Kumar, S.: Deepmala: the unique solvability conditions for a new class of absolute value equation. Yugosl. J. Oper. Res. 33(3), 425\u2013434 (2023). https:\/\/doi.org\/10.2298\/YJOR220515036K","journal-title":"Yugosl. J. Oper. Res."},{"issue":"8","key":"717_CR58","doi-asserted-by":"publisher","first-page":"8912","DOI":"10.3934\/math.2021517","volume":"6","author":"H Zhou","year":"2021","unstructured":"Zhou, H., Wu, S.: On the unique solution of a class of absolute value equations $$Ax-B|Cx|=d$$. AIMS Math. 6(8), 8912\u20138919 (2021). https:\/\/doi.org\/10.3934\/math.2021517","journal-title":"AIMS Math."},{"key":"717_CR59","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/j.aml.2019.01.024","volume":"92","author":"B Huang","year":"2019","unstructured":"Huang, B., Ma, C.: Convergent conditions of the generalized Newton method for absolute value equation over second order cones. Appl. Math. Lett. 92, 151\u2013157 (2019)","journal-title":"Appl. Math. Lett."},{"key":"717_CR60","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1016\/j.apnum.2017.04.012","volume":"120","author":"X-H Miao","year":"2017","unstructured":"Miao, X.-H., Yang, J.-T., Saheya, B., Chen, J.-S.: A smoothing Newton method for absolute value equation associated with second-order cone. Appl. Numer. Math. 120, 82\u201396 (2017)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"717_CR61","doi-asserted-by":"publisher","first-page":"47","DOI":"10.3934\/naco.2021050","volume":"12","author":"X-H Miao","year":"2022","unstructured":"Miao, X.-H., Yao, K., Yang, C.-Y., Chen, J.-S.: Levenberg-Marquardt method for absolute value equation associated with second-order cone. Num. Algebra Control Optim. 12(1), 47\u201361 (2022)","journal-title":"Num. Algebra Control Optim."},{"key":"717_CR62","doi-asserted-by":"publisher","first-page":"206","DOI":"10.1016\/j.apnum.2018.08.019","volume":"135","author":"CT Nguyen","year":"2019","unstructured":"Nguyen, C.T., Saheya, B., Chang, Y.-L., Chen, J.-S.: Unified smoothing functions for absolute value equation associated with second-order cone. Appl. Numer. Math. 135, 206\u2013227 (2019)","journal-title":"Appl. Numer. Math."},{"issue":"6","key":"717_CR63","doi-asserted-by":"publisher","first-page":"3157","DOI":"10.1007\/s40840-022-01370-5","volume":"45","author":"FPA Beik","year":"2022","unstructured":"Beik, F.P.A., Najafi-Kalyani, M., Mollahasani, S.: On the solvability of tensor absolute value equations. Bull. Malays. Math. Sci. Soc. 45(6), 3157\u20133176 (2022). https:\/\/doi.org\/10.1007\/s40840-022-01370-5","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"717_CR64","doi-asserted-by":"publisher","first-page":"108262","DOI":"10.1016\/j.aml.2022.108262","volume":"133","author":"L-B Cui","year":"2022","unstructured":"Cui, L.-B., Lian, G.-G., Yuan, J.-Y.: Existence and uniqueness of positive solution for multilinear systems with generalized strong M-tensor. Appl. Math. Lett. 133, 108262 (2022). https:\/\/doi.org\/10.1016\/j.aml.2022.108262","journal-title":"Appl. Math. Lett."},{"issue":"9","key":"717_CR65","doi-asserted-by":"publisher","first-page":"1695","DOI":"10.1007\/s11425-017-9238-6","volume":"61","author":"S Du","year":"2018","unstructured":"Du, S., Zhang, L., Chen, C., Qi, L.: Tensor absolute value equations. Sci China Math 61(9), 1695\u20131710 (2018)","journal-title":"Sci China Math"},{"key":"717_CR66","doi-asserted-by":"publisher","first-page":"109479","DOI":"10.1016\/j.aml.2025.109479","volume":"164","author":"T Luo","year":"2025","unstructured":"Luo, T., Liu, J., Chen, C., Wang, Q.: A monotone block coordinate descent method for solving absolute value equations. Appl. Math. Lett. 164, 109479 (2025). https:\/\/doi.org\/10.1016\/j.aml.2025.109479","journal-title":"Appl. Math. Lett."},{"issue":"7","key":"717_CR67","doi-asserted-by":"publisher","first-page":"1527","DOI":"10.1007\/s11590-011-0347-6","volume":"6","author":"OL Mangasarian","year":"2012","unstructured":"Mangasarian, O.L.: Primal-dual bilinear programming solution of the absolute value equation. Optim. Lett. 6(7), 1527\u20131533 (2012)","journal-title":"Optim. Lett."},{"issue":"4","key":"717_CR68","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1007\/s11590-012-0469-5","volume":"7","author":"OL Mangasarian","year":"2013","unstructured":"Mangasarian, O.L.: Absolute value equation solution via dual complementarity. Optim. Lett. 7(4), 625\u2013630 (2013)","journal-title":"Optim. Lett."},{"issue":"5","key":"717_CR69","doi-asserted-by":"publisher","first-page":"1027","DOI":"10.1007\/s11590-011-0332-0","volume":"6","author":"MA Noor","year":"2012","unstructured":"Noor, M.A., Iqbal, J., Noor, K.I., Al-Said, E.: On an iterative method for solving absolute value equations. Optim. Lett. 6(5), 1027\u20131033 (2012)","journal-title":"Optim. Lett."},{"issue":"3","key":"717_CR70","first-page":"449","volume":"11","author":"S Shahsavari","year":"2021","unstructured":"Shahsavari, S., Ketabchi, S.: The proximal methods for solving absolute value equation. Num. Algebra Control and Optim. 11(3), 449\u2013460 (2021)","journal-title":"Num. Algebra Control and Optim."},{"issue":"1","key":"717_CR71","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s11590-008-0094-5","volume":"3","author":"OL Mangasarian","year":"2009","unstructured":"Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101\u2013108 (2009)","journal-title":"Optim. Lett."},{"key":"717_CR72","doi-asserted-by":"publisher","DOI":"10.1007\/s11590-024-02121-0","author":"C-H Guo","year":"2025","unstructured":"Guo, C.-H.: Comments on finite termination of the generalized Newton method for absolute value equations. Optim. Lett. (2025). https:\/\/doi.org\/10.1007\/s11590-024-02121-0","journal-title":"Optim. Lett."},{"key":"717_CR73","doi-asserted-by":"publisher","first-page":"1663","DOI":"10.1007\/s11590-021-01837-7","volume":"16","author":"M Radons","year":"2022","unstructured":"Radons, M., Rump, S.M.: Convergence results for some piecewise linear solvers. Optim. Lett. 16, 1663\u20131673 (2022). https:\/\/doi.org\/10.1007\/s11590-021-01837-7","journal-title":"Optim. Lett."},{"issue":"1","key":"717_CR74","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/s10589-009-9242-9","volume":"48","author":"L Caccetta","year":"2011","unstructured":"Caccetta, L., Qu, B., Zhou, G.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48(1), 45\u201358 (2011)","journal-title":"Comput. Optim. Appl."},{"issue":"1","key":"717_CR75","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/s10589-016-9837-x","volume":"65","author":"JYB Cruz","year":"2016","unstructured":"Cruz, J.Y.B., Ferreira, O.P., Prudente, L.F.: On the global convergence of the inexact semi-smooth Newton method for absolute value equation. Comput. Optim. Appl. 65(1), 93\u2013108 (2016)","journal-title":"Comput. Optim. Appl."},{"key":"717_CR76","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1023\/A:1021902825707","volume":"41","author":"LO Jay","year":"2001","unstructured":"Jay, L.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422\u2013429 (2001)","journal-title":"BIT Numer. Math."},{"issue":"1","key":"717_CR77","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1023\/A:1014861331301","volume":"113","author":"L Qi","year":"2002","unstructured":"Qi, L., Sun, D.: Smoothing functions and smoothing Newton method for complementarity and variational inequality problems. J. Optim. Theory Appl. 113(1), 121\u2013147 (2002)","journal-title":"J. Optim. Theory Appl."},{"issue":"1","key":"717_CR78","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1007\/s12190-016-1065-0","volume":"56","author":"B Saheya","year":"2018","unstructured":"Saheya, B., Yu, C.-H., Chen, J.-S.: Numerical comparisons based on four smoothing functions for absolute value equation. J. Appl. Math. Comput. 56(1), 131\u2013149 (2018)","journal-title":"J. Appl. Math. Comput."},{"issue":"4","key":"717_CR79","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1016\/j.orl.2019.03.014","volume":"47","author":"J Tang","year":"2019","unstructured":"Tang, J., Zhou, J.: A quadratically convergent descent method for the absolute value equation $$Ax+ B| x|= b$$. Oper. Res. Lett. 47(4), 229\u2013234 (2019)","journal-title":"Oper. Res. Lett."},{"issue":"2","key":"717_CR80","doi-asserted-by":"publisher","first-page":"1258","DOI":"10.3934\/math.2021078","volume":"6","author":"Y Cao","year":"2021","unstructured":"Cao, Y., Shi, Q., Zhu, S.-L.: A relaxed generalized Newton iteration method for generalized absolute value equations. AIMS Math. 6(2), 1258\u20131275 (2021)","journal-title":"AIMS Math."},{"issue":"1","key":"717_CR81","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s40064-016-2720-5","volume":"5","author":"J Feng","year":"2016","unstructured":"Feng, J., Liu, S.: An improved generalized Newton method for absolute value equations. Springerplus 5(1), 1\u201310 (2016)","journal-title":"Springerplus"},{"issue":"3","key":"717_CR82","doi-asserted-by":"publisher","first-page":"1055","DOI":"10.1007\/s10957-016-0956-4","volume":"170","author":"C-X Li","year":"2016","unstructured":"Li, C.-X.: A modified generalized Newton method for absolute value equations. J. Optim. Theory Appl. 170(3), 1055\u20131059 (2016)","journal-title":"J. Optim. Theory Appl."},{"key":"717_CR83","first-page":"1","volume":"1","author":"C-X Li","year":"2016","unstructured":"Li, C.-X.: On the modified Hermitian and skew-Hermitian splitting iteration methods for a class of weakly absolute value equations. J. Inequ. Appl. 1, 1\u201310 (2016)","journal-title":"J. Inequ. Appl."},{"issue":"2","key":"717_CR84","doi-asserted-by":"publisher","first-page":"1743","DOI":"10.3934\/math.2021104","volume":"6","author":"S-X Miao","year":"2021","unstructured":"Miao, S.-X., Xiong, X.-T., Wen, J.: On Picard-SHSS iteration method for absolute value equation. AIMS Math. 6(2), 1743\u20131753 (2021)","journal-title":"AIMS Math."},{"issue":"8","key":"717_CR85","doi-asserted-by":"publisher","first-page":"2191","DOI":"10.1007\/s11590-014-0727-9","volume":"8","author":"DK Salkuyeh","year":"2014","unstructured":"Salkuyeh, D.K.: The Picard-HSS iteration method for absolute value equations. Optim. Lett. 8(8), 2191\u20132202 (2014)","journal-title":"Optim. Lett."},{"key":"717_CR86","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1016\/j.amc.2015.05.018","volume":"265","author":"J-J Zhang","year":"2015","unstructured":"Zhang, J.-J.: The relaxed nonlinear PHSS-like iteration method for absolute value equations. Appl. Math. Comput. 265, 266\u2013274 (2015)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"717_CR87","first-page":"1","volume":"48","author":"M-Z Zhu","year":"2018","unstructured":"Zhu, M.-Z., Qi, Y.-E.: The nonlinear HSS-like iterative method for absolute value equations. IAENG Int. J. Appl. Math. 48(3), 1\u20135 (2018)","journal-title":"IAENG Int. J. Appl. Math."},{"key":"717_CR88","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/j.amc.2017.05.035","volume":"311","author":"Y-F Ke","year":"2017","unstructured":"Ke, Y.-F., Ma, C.-F.: SOR-like iteration method for solving absolute value equations. Appl. Math. Comput. 311, 195\u2013202 (2017)","journal-title":"Appl. Math. Comput."},{"key":"717_CR89","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/j.aml.2019.03.033","volume":"97","author":"P Guo","year":"2019","unstructured":"Guo, P., Wu, S.-L., Li, C.-X.: On the SOR-like iteration method for solving absolute value equations. Appl. Math. Lett. 97, 107\u2013113 (2019)","journal-title":"Appl. Math. Lett."},{"key":"717_CR90","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1016\/j.apnum.2020.05.013","volume":"156","author":"X Dong","year":"2020","unstructured":"Dong, X., Shao, X.-H., Shen, H.-L.: A new SOR-like method for solving absolute value equations. Appl. Numer. Math. 156, 410\u2013421 (2020)","journal-title":"Appl. Numer. Math."},{"key":"717_CR91","doi-asserted-by":"publisher","first-page":"105990","DOI":"10.1016\/j.aml.2019.07.021","volume":"99","author":"Y Ke","year":"2020","unstructured":"Ke, Y.: The new iteration algorithm for absolute value equation. Appl. Math. Lett. 99, 105990 (2020)","journal-title":"Appl. Math. Lett."},{"key":"717_CR92","doi-asserted-by":"crossref","unstructured":"Li, C.-X., Wu, S.-L.: Modified SOR-like method for absolute value equations. Mathematical Problems in Engineering (2020)","DOI":"10.1155\/2020\/9231639"},{"issue":"2","key":"717_CR93","doi-asserted-by":"publisher","first-page":"1137","DOI":"10.1007\/s13160-023-00641-3","volume":"41","author":"X-M Lv","year":"2024","unstructured":"Lv, X.-M., Miao, S.-X.: An inexact fixed point iteration method for solving absolute value equation. Japan J. Indust. Appl. Math. 41(2), 1137\u20131148 (2024). https:\/\/doi.org\/10.1007\/s13160-023-00641-3","journal-title":"Japan J. Indust. Appl. Math."},{"issue":"3","key":"717_CR94","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1080\/02331934.2020.1804568","volume":"71","author":"D Yu","year":"2022","unstructured":"Yu, D., Chen, C., Han, D.: A modified fixed point iteration method for solving the system of absolute value equations. Optimization 71(3), 449\u2013461 (2022)","journal-title":"Optimization"},{"issue":"1","key":"717_CR95","doi-asserted-by":"publisher","first-page":"258","DOI":"10.1186\/s13660-020-02525-3","volume":"2020","author":"L Zheng","year":"2020","unstructured":"Zheng, L.: The Picard-HSS-SOR iteration method for absolute value equations. J. Inequ. Appl. 2020(1), 258 (2020)","journal-title":"J. Inequ. Appl."},{"key":"717_CR96","doi-asserted-by":"publisher","first-page":"113578","DOI":"10.1016\/j.cam.2021.113578","volume":"394","author":"H-Y Zhou","year":"2021","unstructured":"Zhou, H.-Y., Wu, S.-L., Li, C.-X.: Newton-based matrix splitting method for generalized absolute value equation. J. Comput. Appl. Math. 394, 113578 (2021)","journal-title":"J. Comput. Appl. Math."},{"key":"717_CR97","doi-asserted-by":"publisher","first-page":"156","DOI":"10.1016\/j.amc.2016.08.020","volume":"293","author":"V Edalatpour","year":"2017","unstructured":"Edalatpour, V., Hezari, D., Salkuyeh, D.K.: A generalization of the Gauss-Seidel iteration method for solving absolute value equations. Appl. Math. Comput. 293, 156\u2013167 (2017)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"717_CR98","first-page":"1","volume":"15","author":"A Fakharzadeh","year":"2021","unstructured":"Fakharzadeh, A., Shams, N.N.: An efficient algorithm for solving absolute value equations. J. Math. Exten. 15(3), 1\u201323 (2021)","journal-title":"J. Math. Exten."},{"key":"717_CR99","doi-asserted-by":"publisher","first-page":"40","DOI":"10.2306\/scienceasia1513-1874.2018.44.040","volume":"44","author":"J He","year":"2018","unstructured":"He, J., Liu, Y., Tian, J.: Two numerical iteration methods for solving absolute value equations. Sci. Asia 44, 40\u201345 (2018)","journal-title":"Sci. Asia"},{"key":"717_CR100","doi-asserted-by":"publisher","first-page":"3643","DOI":"10.22436\/jnsa.010.07.24","volume":"10","author":"C-Q Lv","year":"2017","unstructured":"Lv, C.-Q., Ma, C.-F.: Picard splitting method and Picard CG method for solving the absolute value equation. J. Nonlinear Sci. Appl 10, 3643\u20133654 (2017)","journal-title":"J. Nonlinear Sci. Appl"},{"issue":"12","key":"717_CR101","doi-asserted-by":"publisher","first-page":"4171","DOI":"10.2298\/FIL2012171S","volume":"34","author":"NN Shams","year":"2020","unstructured":"Shams, N.N., Jahromi, A.F., Beik, F.P.A.: Iterative schemes induced by block splittings for solving absolute value equations. Filomat 34(12), 4171\u20134188 (2020)","journal-title":"Filomat"},{"issue":"5","key":"717_CR102","doi-asserted-by":"publisher","first-page":"5171","DOI":"10.3934\/math.2020332","volume":"5","author":"S Wu","year":"2020","unstructured":"Wu, S., Li, C.: A special shift splitting iteration method for absolute value equation. AIMS Math. 5(5), 5171\u20135183 (2020)","journal-title":"AIMS Math."},{"key":"717_CR103","unstructured":"Yu, D., Chen, C., Han, D.: A class of inexact modified Newton-type iteration methods for solving the generalized absolute value equations. preprint, arXiv:2103.10129 (2021)"},{"key":"717_CR104","doi-asserted-by":"publisher","first-page":"589","DOI":"10.13001\/1081-3810.1332","volume":"18","author":"J Rohn","year":"2009","unstructured":"Rohn, J.: An algorithm for solving the absolute value equation. Electron. J. Linear Algebra 18, 589\u2013599 (2009). https:\/\/doi.org\/10.13001\/1081-3810.1332","journal-title":"Electron. J. Linear Algebra"},{"key":"717_CR105","unstructured":"Rohn, J.: An algorithm for solving the absolute value equation: An improvement. Technical report V-1063, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2010). http:\/\/hdl.handle.net\/11104\/0181476"},{"key":"717_CR106","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1016\/j.tcs.2016.02.009","volume":"626","author":"M Radons","year":"2016","unstructured":"Radons, M.: Direct solution of piecewise linear systems. Theor. Comput. Sci. 626, 97\u2013109 (2016). https:\/\/doi.org\/10.1016\/j.tcs.2016.02.009","journal-title":"Theor. Comput. Sci."},{"issue":"3","key":"717_CR107","first-page":"391","volume":"65","author":"A Mansoori","year":"2018","unstructured":"Mansoori, A., Eshaghnezhad, M., Effati, S.: An efficient neural network model for solving the absolute value equations. IEEE Trans. Circuits Syst. II Express Briefs 65(3), 391\u2013395 (2018)","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"issue":"7","key":"717_CR108","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.1007\/s13370-014-0281-8","volume":"26","author":"H Moosaei","year":"2015","unstructured":"Moosaei, H., Ketabchi, S., Jafari, H.: Minimum norm solution of the absolute value equations via simulated annealing algorithm. Afr. Mat. 26(7), 1221\u20131228 (2015)","journal-title":"Afr. Mat."},{"key":"717_CR109","doi-asserted-by":"publisher","first-page":"696","DOI":"10.1016\/j.amc.2015.06.072","volume":"268","author":"H Moosaei","year":"2015","unstructured":"Moosaei, H., Ketabchi, S., Noor, M.A., Iqbal, J., Hooshyarbakhsh, V.: Some techniques for solving absolute value equations. Appl. Math. Comput. 268, 696\u2013705 (2015)","journal-title":"Appl. Math. Comput."},{"key":"717_CR110","doi-asserted-by":"publisher","first-page":"134","DOI":"10.1016\/j.cam.2014.11.062","volume":"282","author":"J Iqbal","year":"2015","unstructured":"Iqbal, J., Iqbal, A., Arif, M.: Levenberg-Marquardt method for solving systems of absolute value equations. J. Comput. Appl. Math. 282, 134\u2013138 (2015)","journal-title":"J. Comput. Appl. Math."},{"key":"717_CR111","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1016\/j.cor.2016.04.005","volume":"87","author":"T Pham Dinh","year":"2017","unstructured":"Pham Dinh, T., Ho, V.T., Le Thi, H.A.: DC programming and DCA for solving Brugnano-Casulli piecewise linear systems. Comput. Oper. Res. 87, 196\u2013204 (2017). https:\/\/doi.org\/10.1016\/j.cor.2016.04.005","journal-title":"Comput. Oper. Res."},{"issue":"5","key":"717_CR112","first-page":"1171","volume":"9","author":"A Wang","year":"2011","unstructured":"Wang, A., Wang, H., Deng, Y.: Interval algorithm for absolute value equations. Central Eur. J. Math. 9(5), 1171\u20131184 (2011)","journal-title":"Central Eur. J. Math."},{"issue":"1","key":"717_CR113","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1007\/s10957-018-1439-6","volume":"181","author":"A Wang","year":"2019","unstructured":"Wang, A., Cao, Y., Chen, J.-X.: Modified Newton-type iteration methods for generalized absolute value equations. J. Optim. Theory Appl. 181(1), 216\u2013230 (2019)","journal-title":"J. Optim. Theory Appl."},{"key":"717_CR114","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1016\/j.apnum.2020.08.001","volume":"158","author":"M Dehghan","year":"2020","unstructured":"Dehghan, M., Shirilord, A.: Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation. Appl. Numer. Math. 158, 425\u2013438 (2020)","journal-title":"Appl. Numer. Math."},{"issue":"5","key":"717_CR115","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1007\/s11590-011-0305-3","volume":"6","author":"J Rohn","year":"2012","unstructured":"Rohn, J.: An algorithm for computing all solutions of an absolute value equation. Optim. Lett. 6(5), 851\u2013856 (2012)","journal-title":"Optim. Lett."},{"key":"717_CR116","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.cam.2017.09.032","volume":"333","author":"A Mansoori","year":"2018","unstructured":"Mansoori, A., Erfanian, M.: A dynamic model to solve the absolute value equations. J. Comput. Appl. Math. 333, 28\u201335 (2018)","journal-title":"J. Comput. Appl. Math."},{"key":"717_CR117","doi-asserted-by":"crossref","unstructured":"Yong, L., Zhang, S., Xiong, W., et al: Feasible interior point method for absolute value equation. In: 2011 Fourth International Conference on Information and Computing, pp. 256\u2013259 (2011). IEEE","DOI":"10.1109\/ICIC.2011.65"},{"key":"717_CR118","first-page":"1","volume":"2018","author":"M Achache","year":"2018","unstructured":"Achache, M., Hazzam, N.: Solving absolute value equations via complementarity and interior point methods. J. Nonlinear Funct. Anal. 2018, 1\u201310 (2018)","journal-title":"J. Nonlinear Funct. Anal."},{"issue":"1","key":"717_CR119","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/j.orl.2022.11.009","volume":"51","author":"S Yang","year":"2023","unstructured":"Yang, S., Wu, S.-L.: A modified Barzilai-Borwein algorithm for the generalized absolute value equation. Oper. Res. Lett. 51(1), 21\u201325 (2023)","journal-title":"Oper. Res. Lett."},{"key":"717_CR120","first-page":"208","volume-title":"Funct. Anal. Optim. Math. Econ.","author":"JB Rosen","year":"1990","unstructured":"Rosen, J.B.: Minimum norm solution to the linear complementarity problem. In: Leifman, L.J. (ed.) Funct. Anal. Optim. Math. Econ., pp. 208\u2013216. Oxford University Press, New York (1990)"},{"issue":"3","key":"717_CR121","doi-asserted-by":"publisher","first-page":"1080","DOI":"10.1007\/s10957-012-0044-3","volume":"154","author":"S Ketabchi","year":"2012","unstructured":"Ketabchi, S., Moosaei, H.: Minimum norm solution to the absolute value equation in the convex case. J. Optim. Theory Appl. 154(3), 1080\u20131087 (2012)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"717_CR122","first-page":"57","volume":"7","author":"S Ketabchi","year":"2017","unstructured":"Ketabchi, S., Moosaei, H.: Augmented Lagrangian method for finding minimum norm solution to the absolute value equation. Iran. J. Num. Anal. Optim. 7(2), 57\u201364 (2017)","journal-title":"Iran. J. Num. Anal. Optim."},{"issue":"1","key":"717_CR123","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/s12209-016-2640-z","volume":"22","author":"X Liu","year":"2016","unstructured":"Liu, X., Fan, J., Li, W.: Concave minimization for sparse solutions of absolute value equations. Trans. Tianjin Univ. 22(1), 89\u201394 (2016)","journal-title":"Trans. Tianjin Univ."},{"issue":"1\u20132","key":"717_CR124","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1007\/s10107-015-0950-x","volume":"159","author":"X Chen","year":"2016","unstructured":"Chen, X., Xiang, S.: Sparse solutions of linear complementarity problems. Math. Program. 159(1\u20132), 539\u2013556 (2016). https:\/\/doi.org\/10.1007\/s10107-015-0950-x","journal-title":"Math. Program."},{"issue":"6","key":"717_CR125","doi-asserted-by":"publisher","first-page":"1882","DOI":"10.1016\/j.camwa.2012.03.015","volume":"64","author":"S Ketabchi","year":"2012","unstructured":"Ketabchi, S., Moosaei, H.: An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side. Comput. Math. Appl. 64(6), 1882\u20131885 (2012)","journal-title":"Comput. Math. Appl."},{"issue":"9\u201310","key":"717_CR126","doi-asserted-by":"publisher","first-page":"2339","DOI":"10.1016\/j.mcm.2011.11.068","volume":"57","author":"S Ketabchi","year":"2013","unstructured":"Ketabchi, S., Moosaei, H., Fallahi, S.: Optimal error correction of the absolute value equation using a genetic algorithm. Math. Comput. Model. 57(9\u201310), 2339\u20132342 (2013)","journal-title":"Math. Comput. Model."},{"issue":"3","key":"717_CR127","doi-asserted-by":"publisher","first-page":"645","DOI":"10.1007\/s10898-020-00948-2","volume":"79","author":"H Moosaei","year":"2021","unstructured":"Moosaei, H., Ketabchi, S., Hlad\u00edk, M.: Optimal correction of the absolute value equations. J. Global Optim. 79(3), 645\u2013667 (2021)","journal-title":"J. Global Optim."},{"issue":"8","key":"717_CR128","doi-asserted-by":"publisher","first-page":"1531","DOI":"10.1080\/02331934.2016.1154963","volume":"65","author":"H Moosaei","year":"2016","unstructured":"Moosaei, H., Ketabchi, S., Pardalos, P.M.: Tikhonov regularization for infeasible absolute value equations. Optimization 65(8), 1531\u20131537 (2016)","journal-title":"Optimization"},{"key":"717_CR129","doi-asserted-by":"publisher","unstructured":"Hashemi, F., Ketabchi, S.: Optimal correction of infeasible equations system as $$Ax+ B| x|= b$$ using $$\\ell _p$$-norm regularization. Boletim da Sociedade Paranaense de Matem\u00e1tica 40 (2022) https:\/\/doi.org\/10.5269\/bspm.44437. in press","DOI":"10.5269\/bspm.44437"},{"issue":"1","key":"717_CR130","doi-asserted-by":"publisher","first-page":"13","DOI":"10.3934\/naco.2019029","volume":"10","author":"F Hashemi","year":"2020","unstructured":"Hashemi, F., Ketabchi, S.: Numerical comparisons of smoothing functions for optimal correction of an infeasible system of absolute value equations. Num. Algebra Control Optim. 10(1), 13 (2020)","journal-title":"Num. Algebra Control Optim."},{"key":"717_CR131","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1017\/S096249291000005X","volume":"19","author":"SM Rump","year":"2010","unstructured":"Rump, S.M.: Verification methods: rigorous results using floating-point arithmetic. Acta Numer 19, 287\u2013449 (2010). https:\/\/doi.org\/10.1017\/S096249291000005X","journal-title":"Acta Numer"},{"issue":"1","key":"717_CR132","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/s13348-011-0057-5","volume":"64","author":"H Wang","year":"2013","unstructured":"Wang, H., Liu, H., Cao, S.: A verification method for enclosing solutions of absolute value equations. Collect. Math. 64(1), 17\u201338 (2013). https:\/\/doi.org\/10.1007\/s13348-011-0057-5","journal-title":"Collect. Math."},{"issue":"3","key":"717_CR133","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1007\/s10092-016-0204-1","volume":"54","author":"HJ Wang","year":"2017","unstructured":"Wang, H.J., Cao, D.X., Liu, H., Qiu, L.: Numerical validation for systems of absolute value equations. Calcolo 54(3), 669\u2013683 (2017). https:\/\/doi.org\/10.1007\/s10092-016-0204-1","journal-title":"Calcolo"},{"issue":"1","key":"717_CR134","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1007\/s10107-021-01756-6","volume":"198","author":"M Zamani","year":"2023","unstructured":"Zamani, M., Hlad\u00edk, M.: Error bounds and a condition number for the absolute value equations. Math. Program. 198(1), 85\u2013113 (2023). https:\/\/doi.org\/10.1007\/s10107-021-01756-6","journal-title":"Math. Program."},{"issue":"6","key":"717_CR135","doi-asserted-by":"publisher","first-page":"707","DOI":"10.1080\/03081087.2010.486403","volume":"59","author":"S Karademir","year":"2011","unstructured":"Karademir, S., Prokopyev, O.A.: A short note on solvability of systems of interval linear equations. Linear Multilinear Algebra 59(6), 707\u2013710 (2011). https:\/\/doi.org\/10.1080\/03081087.2010.486403","journal-title":"Linear Multilinear Algebra"},{"issue":"7","key":"717_CR136","doi-asserted-by":"publisher","first-page":"1390","DOI":"10.1080\/03081087.2014.940827","volume":"63","author":"M Xia","year":"2015","unstructured":"Xia, M., Li, W., Li, H.: Farkas-type theorems for interval linear systems. Linear Multilinear Algebra 63(7), 1390\u20131400 (2015). https:\/\/doi.org\/10.1080\/03081087.2014.940827","journal-title":"Linear Multilinear Algebra"},{"key":"717_CR137","volume-title":"Interval Methods for Systems of Equations","author":"A Neumaier","year":"1990","unstructured":"Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)"},{"key":"717_CR138","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1007\/0-387-32698-7_2","volume-title":"Linear Optimization Problems with Inexact Data","author":"J Rohn","year":"2006","unstructured":"Rohn, J.: Solvability of systems of interval linear equations and inequalities. In: Al, M. (ed.) Linear Optimization Problems with Inexact Data, pp. 35\u201377. Springer, New York (2006)"},{"key":"717_CR139","doi-asserted-by":"crossref","unstructured":"Gerlach, W.: Zur L\u00f6sung linearer Ungleichungssysteme bei St\u00f6rung der rechten Seite und der Koeffizientenmatrix. Math. Operationsforsch. Stat., Ser. Optimization 12, 41\u201343 (1981). in German","DOI":"10.1080\/02331938108842705"},{"key":"717_CR140","doi-asserted-by":"publisher","unstructured":"Hlad\u00edk, M., Zamani, M.: Absolute value programming. In: Pardalos, P.M., Prokopyev, O. (eds.) Encyclopedia of Optimization, 3rd edn., pp. 1\u20137. Springer, Cham (2023). https:\/\/doi.org\/10.1007\/978-3-030-54621-2_725-1","DOI":"10.1007\/978-3-030-54621-2_725-1"},{"key":"717_CR141","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1007\/BF01386090","volume":"6","author":"W Oettli","year":"1964","unstructured":"Oettli, W., Prager, W.: Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numer. Math. 6, 405\u2013409 (1964)","journal-title":"Numer. Math."},{"key":"717_CR142","unstructured":"Hlad\u00edk, M., Hartman, D.: Absolute value linear programming. preprint arXiv: 2307.03510 (2023). https:\/\/arxiv.org\/abs\/2307.03510"},{"issue":"2","key":"717_CR143","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1080\/02331934.2011.644289","volume":"63","author":"S Yamanaka","year":"2014","unstructured":"Yamanaka, S., Fukushima, M.: A branch-and-bound method for absolute value programs. Optimization 63(2), 305\u2013319 (2014). https:\/\/doi.org\/10.1080\/02331934.2011.644289","journal-title":"Optimization"},{"key":"717_CR144","unstructured":"Hlad\u00edk, M.: Interval linear programming: A survey. In: Mann, Z.A. (ed.) Linear Programming\u2013 New Frontiers in Theory and Applications, pp. 85\u2013120. Nova Science Publishers, New York (2012). Chap. 2"},{"key":"717_CR145","doi-asserted-by":"crossref","unstructured":"Gabrel, V., Murat, C., Remli, N.: Best and worst optimum for linear programs with interval right hand sides. In: al., H.A. (ed.) Modelling, Computation and Optimization in Information Systems and Management Sciences. Second International Conference MCO 2008, Metz, France. Proceedings. Springer, Berlin (2008)","DOI":"10.1007\/978-3-540-87477-5_14"},{"issue":"5","key":"717_CR146","doi-asserted-by":"publisher","first-page":"1145","DOI":"10.1007\/s11590-019-01385-1","volume":"14","author":"MA Raayatpanah","year":"2020","unstructured":"Raayatpanah, M.A., Moosaei, H., Pardalos, P.M.: Absolute value equations with uncertain data. Optim. Lett. 14(5), 1145\u20131156 (2020). https:\/\/doi.org\/10.1007\/s11590-019-01385-1","journal-title":"Optim. Lett."},{"issue":"7","key":"717_CR147","doi-asserted-by":"publisher","first-page":"3705","DOI":"10.1007\/s00500-025-10655-3","volume":"29","author":"M Hlad\u00edk","year":"2025","unstructured":"Hlad\u00edk, M., Pt\u00e1\u010dkov\u00e1, L.: Absolute value equations with interval uncertainty. Soft. Comput. 29(7), 3705\u20133718 (2025). https:\/\/doi.org\/10.1007\/s00500-025-10655-3","journal-title":"Soft. Comput."},{"issue":"4","key":"717_CR148","doi-asserted-by":"publisher","first-page":"549","DOI":"10.23952\/jnva.7.2023.4.06","volume":"7","author":"Y Lu","year":"2023","unstructured":"Lu, Y., Ma, H.-M., Xue, D.-Y., Chen, J.-S.: Absolute value equations with data uncertainty in the $$l_1$$ and $$l_\\infty $$ norm balls. J. Nonlinear Var. Anal. 7(4), 549\u2013561 (2023). https:\/\/doi.org\/10.23952\/jnva.7.2023.4.06","journal-title":"J. Nonlinear Var. Anal."},{"issue":"1\u20133","key":"717_CR149","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/BF01581088","volume":"57","author":"PM Pardalos","year":"1992","unstructured":"Pardalos, P.M., Vavasis, S.A.: Open questions in complexity theory for numerical optimization. Math. Program. 57(1\u20133), 337\u2013339 (1992). https:\/\/doi.org\/10.1007\/BF01581088","journal-title":"Math. Program."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00717-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-025-00717-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-025-00717-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T16:02:13Z","timestamp":1767974533000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-025-00717-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,8]]},"references-count":149,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["717"],"URL":"https:\/\/doi.org\/10.1007\/s10589-025-00717-5","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,8]]},"assertion":[{"value":"23 October 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 August 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}