The Sparsest Solution to the System of Absolute Value Equations

Original Paper
First Online:
Received:
Revised:
Accepted:

Abstract

On one hand, to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas; and on the other hand, the system of absolute value equations (AVEs) has attracted a lot of attention since many practical problems can be equivalently transformed as a system of AVEs. Motivated by the development of these two aspects, we consider the problem to find the sparsest solution to the system of AVEs in this paper. We first propose the model of the concerned problem, i.e., to find the solution to the system of AVEs with the minimum \(l_0\)-norm. Since \(l_0\)-norm is difficult to handle, we relax the problem into a convex optimization problem and discuss the necessary and sufficient conditions to guarantee the existence of the unique solution to the convex relaxation problem. Then, we prove that under such conditions the unique solution to the convex relaxation is exactly the sparsest solution to the system of AVEs. When the concerned system of AVEs reduces to the system of linear equations, the obtained results reduce to those given in the literature. The theoretical results obtained in this paper provide an important basis for designing numerical method to find the sparsest solution to the system of AVEs.

Keywords

Absolute value equations The sparsest solution Minimum \(l_1\)-norm solution 

Mathematics Subject Classification

62B10 90C26 90C59 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Caccetta, L., Qu, B., Zhou, G.L.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48, 45–58 (2011)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Cai, T.T., Zhang, A.: Sparse representation of a polytope and recovery of sparse signals and low-rank matrices. IEEE Trans. Inf. Theory. 60, 122–132 (2014)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Candès, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory. 51, 4203–4215 (2005)CrossRefMATHGoogle Scholar
  4. 4.
    Elad, M.: Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer, New York (2010)CrossRefGoogle Scholar
  5. 5.
    Eldar, Y.C., Kutyniok, G.: Compressed Sensing: Theory and Applications. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  6. 6.
    Hu, S.L., Huang, Z.H.: A note on absolute value equations. Optim. Lett. 4, 417–424 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    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, 1490–1501 (2011)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Juditsky, A., Nemirovski, A.S.: On verifiable sufficient conditions for sparse signal recovery via \(l_1\) minimization. Math. Program. 127, 57–88 (2011)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Ketabchi, S., Moosaei, H.: Minimum norm solution to the absolute value equation in the convex case. J. Optim. Theory Appl. 154, 1080–1087 (2012)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Kong, L.C., Xiu, N.H.: New bounds for restricted isometry constants in low-rank matrix recovery. Optimization-online, http://www.optimization-online.org/DB_FILE/2011/01/2894 (2011)
  11. 11.
    Kong, L.C., Xiu, N.H.: Exact low-rank matrix recovery via nonconvex schatten \(p\)-minimization. Asia-Pac. J. Oper. Res. 30, 1340010 (2013)CrossRefGoogle Scholar
  12. 12.
    Kong, L.C., Tuncel, L., Xiu, N.H.: \(s\)-goodness for low-rank matrix recovery. Abstr. Appl. Anal. 2013, Article ID: 101974 (2013).Google Scholar
  13. 13.
    Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419, 359–367 (2006)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Mangasarian, O.L.: Absolute value programming. Comput. Optim. Appl. 36, 43–53 (2007)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Mangasarian, O.L.: Absolute value equation solution via dual complementarity. Optim. Lett. 7, 625–630 (2013)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Mangasarian, O.L.: Absolute value equation solution via linear programming. J. Optim. Theory Appl. 161, 870–876 (2014)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Natarajan, B.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Prokopyev, O.A., Butenko, S., Trapp, A.: Checking solvability of systems of interval linear equations and inequalities via mixed integer programming. Eur. J. Oper. Res. 199, 117–121 (2009)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Prokopyev, O.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44, 363–372 (2009)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Rohn, J.: A theorem of the alternatives for the equation \(Ax + B|x| = b\). Linear Algebra Appl. 52, 421–426 (2004)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Tropp, J., Gilbert, A.: Signal recovery from random measurements via orthogonal mathcing pursuit. IEEE Trans. Inf. Theory. 53, 4655–4666 (2007)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Zhao, Y.B.: RSP-Based analysis for sparest and least \(l_1\)-norm solutions to underdetermined linear systems. IEEE Trans. Signal Process. 61, 5777–5788 (2013)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Zhao, Y.B.: Equivalence and strong equivalence between the sparsest and least \(l_1\)-norm nonnegative solutions of linear systems and their applications. J. Oper. Res. Soc. China. 2, 171–193 (2014)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Zhang, C., Wei, Q.J.: Global and finite convergence of a generalized Newton method for absolute value equations. J. Optim. Theory Appl. 143, 391–403 (2009)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Zhang, M., Huang, Z.H., Zhang, Y.: Restricted \(p\)-isometry properties of nonconvex matrix recovery. IEEE Trans. Inf. Theory. 59, 4316–4323 (2013)CrossRefGoogle Scholar
  27. 27.
    Zhang, Y.: Theory of compressive sensing via \(l_1\) minimization: a Non-RIP analysis and extensions. J. Oper. Res. Soc. China. 1, 79–105 (2013)CrossRefMATHGoogle Scholar
  28. 28.
    Zhou, S.L., Kong, L.C., Xiu, N.H.: New bounds for RIC in compressed sensing. J. Oper. Res. Soc. China. 1, 227–237 (2013)CrossRefMATHGoogle Scholar

Copyright information

© Operations Research Society of China, Periodicals Agency of Shanghai University, and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceTianjin UniversityTianjinChina
  2. 2.Center for Applied MathematicsTianjin UniversityTianjinChina

Personalised recommendations