Bounds for the solutions of absolute value equations. (English) Zbl 1411.90322
Summary: In the recent years, there has been an intensive research of absolute value equations \(Ax-b=B|x|\). Various methods were developed, but less attention has been paid to approximating or bounding the solutions. We start filling this gap by proposing several outer approximations of the solution set. We present conditions for unsolvability and for existence of exponentially many solutions, too, and compare them with the known conditions. Eventually, we carried out numerical experiments to compare the methods with respect to computational time and quality of estimation. This helps in identifying the cases, in which the bounds are tight enough to determine the signs of the solution, and therefore also the solution itself.
MSC:
| 90C30 | Nonlinear programming |
| 90C59 | Approximation methods and heuristics in mathematical programming |
| 65G40 | General methods in interval analysis |
| 15A06 | Linear equations (linear algebraic aspects) |
References:
| [1] | Baharev, A., Achterberg, T., Rév, E.: Computation of an extractive distillation column with affine arithmetic. AIChE J. 55(7), 1695-1704 (2009) · doi:10.1002/aic.11777 |
| [2] | Bello Cruz, J.Y., 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), 93-108 (2016) · Zbl 1353.90155 · doi:10.1007/s10589-016-9837-x |
| [3] | Caccetta, L., Qu, B., Zhou, G.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48(1), 45-58 (2011) · Zbl 1230.90195 · doi:10.1007/s10589-009-9242-9 |
| [4] | Esmaeili, H., Mahmoodabadi, E., Ahmadi, M.: A uniform approximation method to solve absolute value equation. Bull. Iran. Math. Soc. 41(5), 1259-1269 (2015) · Zbl 1402.65044 |
| [5] | Fiedler, M., Nedoma, J., Ramík, J., Rohn, J., Zimmermann, K.: Linear Optimization Problems with Inexact Data. Springer, New York (2006) · Zbl 1106.90051 |
| [6] | Haghani, F.K.: On generalized Traub’s method for absolute value equations. J. Optim. Theory Appl. 166(2), 619-625 (2015) · Zbl 1391.65106 · doi:10.1007/s10957-015-0712-1 |
| [7] | Hladík, M.: New operator and method for solving real preconditioned interval linear equations. SIAM J. Numer. Anal. 52(1), 194-206 (2014) · Zbl 1293.65043 · doi:10.1137/130914358 |
| [8] | Hu, S., Huang, Z.: A note on absolute value equations. Optim. Lett. 4(3), 417-424 (2010) · Zbl 1202.90251 · doi:10.1007/s11590-009-0169-y |
| [9] | Jiang, X., Zhang, Y.: A smoothing-type algorithm for absolute value equations. J. Ind. Manag. Optim. 9(4), 789-798 (2013) · Zbl 1281.90023 · doi:10.3934/jimo.2013.9.789 |
| [10] | 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), 1882-1885 (2012) · Zbl 1268.15011 · doi:10.1016/j.camwa.2012.03.015 |
| [11] | Mangasarian, O.L.: Absolute value programming. Comput. Optim. Appl. 36(1), 43-53 (2007) · Zbl 1278.90386 · doi:10.1007/s10589-006-0395-5 |
| [12] | Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101-108 (2009) · Zbl 1154.90599 · doi:10.1007/s11590-008-0094-5 |
| [13] | Mangasarian, O.L.: Absolute value equation solution via linear programming. J. Optim. Theory Appl. 161(3), 870-876 (2014) · Zbl 1293.90041 · doi:10.1007/s10957-013-0461-y |
| [14] | Mangasarian, O.L.: Linear complementarity as absolute value equation solution. Optim. Lett. 8(4), 1529-1534 (2014) · Zbl 1288.90109 · doi:10.1007/s11590-013-0656-z |
| [15] | Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419(2), 359-367 (2006) · Zbl 1172.15302 · doi:10.1016/j.laa.2006.05.004 |
| [16] | Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000) · Zbl 0962.15001 · doi:10.1137/1.9780898719512 |
| [17] | Moosaei, H., Ketabchi, S., Noor, M.A., Iqbal, J., Hooshyarbakhsh, V.: Some techniques for solving absolute value equations. Appl. Math. Comput. 268, 696-705 (2015) · Zbl 1410.65192 |
| [18] | Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990) · Zbl 0715.65030 |
| [19] | Ning, S., Kearfott, R.B.: A comparison of some methods for solving linear interval equations. SIAM J. Numer. Anal. 34(4), 1289-1305 (1997) · Zbl 0889.65022 · doi:10.1137/S0036142994270995 |
| [20] | Noor, M.A., Iqbal, J., Noor, K.I., Al-Said, E.A.: On an iterative method for solving absolute value equations. Optim. Lett. 6(5), 1027-1033 (2012) · Zbl 1254.90149 · doi:10.1007/s11590-011-0332-0 |
| [21] | Prokopyev, O.A.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44(3), 363-372 (2009) · Zbl 1181.90263 · doi:10.1007/s10589-007-9158-1 |
| [22] | Rohn, J.: Cheap and tight bounds: the recent result by E. Hansen can be made more efficient. Interval Comput. 1993(4), 13-21 (1993) · Zbl 0830.65019 |
| [23] | Rohn, J.: On unique solvability of the absolute value equation. Optim. Lett. 3(4), 603-606 (2009) · Zbl 1172.90009 · doi:10.1007/s11590-009-0129-6 |
| [24] | Rohn, J.: An improvement of the Bauer-Skeel bounds. Technical report V-1065, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2010). http://uivtx.cs.cas.cz/ rohn/publist/bauerskeel.pdf · Zbl 1343.65060 |
| [25] | Rohn, J.: An algorithm for computing all solutions of an absolute value equation. Optim. Lett. 6(5), 851-856 (2012) · Zbl 1254.90260 · doi:10.1007/s11590-011-0305-3 |
| [26] | Rohn, J.: A manual of results on interval linear problems. Technical Report 1164, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2012). http://www.library.sk/arl-cav/en/detail/?&idx=cav_un_epca*0381706 · Zbl 1293.65043 |
| [27] | 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). http://www.library.sk/arl-cav/sk/detail-cav_un_epca-0424937-/ · Zbl 1402.65044 |
| [28] | Rohn, J., Hooshyarbakhsh, V., Farhadsefat, R.: An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optim. Lett. 8(1), 35-44 (2014) · Zbl 1316.90052 · doi:10.1007/s11590-012-0560-y |
| [29] | Wang, A., Wang, H., Deng, Y.: Interval algorithm for absolute value equations. Cent. Eur. J. Math. 9(5), 1171-1184 (2011) · Zbl 1236.65047 · doi:10.2478/s11533-011-0067-2 |
| [30] | Wu, S., Guo, P.: On the unique solvability of the absolute value equation. J. Optim. Theory Appl. 169(2), 705-712 (2016) · Zbl 1343.65060 · doi:10.1007/s10957-015-0845-2 |
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