Skip to main content
SpringerLink
Log in

On the Solution Uniqueness Characterization in the L1 Norm and Polyhedral Gauge Recovery

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper first proposes another proof of the necessary and sufficient conditions of solution uniqueness in 1-norm minimization given recently by H. Zhang, W. Yin, and L. Cheng. The analysis avoids the need of the surjectivity assumption made by these authors and should be mainly appealing by its short length (it can therefore be proposed to students exercising in convex optimization). In the second part of the paper, the previous existence and uniqueness characterization is extended to the recovery problem where the L1 norm is substituted by a polyhedral gauge. In addition to present interest for a number of practical problems, this extension clarifies the geometrical aspect of the previous uniqueness characterization. Numerical techniques are proposed to compute a solution to the polyhedral gauge recovery problem in polynomial time and to check its possible uniqueness by a simple linear algebra test.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Zhang, H., Yin, W., Cheng, L.: Necessary and sufficient conditions of solution uniqueness in 1-norm minimization. J. Optim. Theory Appl. 164(1), 109–122 (2015). doi:10.1007/s10957-014-0581-z

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, S., Donoho, D., Saunders, M.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20, 33–61 (1998). doi:10.1137/S1064827596304010

    Article  MathSciNet  MATH  Google Scholar 

  3. Natarajan, B.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24(2), 227–234 (2005). doi:10.1137/S0097539792240406

    Article  MathSciNet  MATH  Google Scholar 

  4. Candès, E., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(11), 4203–4215 (2005). doi:10.1109/TIT.2005.858979

    Article  MathSciNet  MATH  Google Scholar 

  5. Donoho, D.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). doi:10.1109/TIT.2006.871582

    Article  MathSciNet  MATH  Google Scholar 

  6. Blanchard, J., Cartis, C., Tanner, J.: Compressed sensing: how sharp is the restricted isometry property? SIAM Rev. 53(1), 105–125 (2011). doi:10.1137/090748160

    Article  MathSciNet  MATH  Google Scholar 

  7. Eldar, Y., Kutyniok, G. (eds.): Compressed Sensing: Theory and Applications. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  8. Harchaoui, Z., Juditsky, A., Nemirovski, A.: Conditional gradient algorithms for norm-regularized smooth convex optimization. Mathematical Programming (2014). doi:10.1007/s10107-014-0778-9

  9. Juditsky, A., Kılınç Karzan, F., Nemirovski, A.: Verifiable conditions of \(\ell _1\)-recovery for sparse signals with sign restrictions. Math. Program. 127(1), 89–122 (2011). doi:10.1007/s10107-010-0418-y

    Article  MathSciNet  MATH  Google Scholar 

  10. Juditsky, A., Nemirovski, A.: Accuracy guarantees for \(\ell _1\)-recovery. IEEE Trans. Inf. Theory 57(12), 7818–7839 (2011). doi:10.1109/TIT.2011.2162569

    Article  MathSciNet  Google Scholar 

  11. Juditsky, A., Nemirovski, A.: On verifiable sufficient conditions for sparse signal recovery via \(\ell _1\) minimization. Math. Program. 127(1), 57–88 (2011). doi:10.1007/s10107-010-0417-z

    Article  MathSciNet  MATH  Google Scholar 

  12. d’Aspremont, A., El Ghaoui, L.: Testing the nullspace property using semidefinite programming. Math. Program. 127(1), 123–144 (2011). doi:10.1007/s10107-010-0416-0

    Article  MathSciNet  MATH  Google Scholar 

  13. Nesterov, Y., Nemirovski, A.: On first-order algorithms for l1/nuclear norm minimization. Acta Numerica 2013(22), 509–575 (2013). doi:10.1017/S096249291300007X

    Article  MATH  Google Scholar 

  14. Donoho, D., Huo, X.: Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory 47(7), 2845–2862 (2001). doi:10.1109/18.959265

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhang, Y.: A simple proof for recoverability of \(\ell _1\)-minimization: go over or under? Technical Report TR05-09, Department of Computational and Applied Mathematics, Rice University, P.O. Box: Houston. Texas 77251(2005), (1892)

  16. Candès, E., Tao, T.: Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006). doi:10.1109/TIT.2006.885507

    Article  MathSciNet  MATH  Google Scholar 

  17. Cohen, A., Dahmen, W., DeVore, R.: Compressed sensing and best k-term approximation. J. Am. Math. Soc. 22(1), 211–231 (2009). doi:10.1090/S0894-0347-08-00610-3

    Article  MathSciNet  MATH  Google Scholar 

  18. Chandrasekaran, V., Recht, B., Parrilo, P., Willsky, A.: The convex geometry of linear inverse problems. Found. Comput. Math. 12(6), 805–849 (2012). doi:10.1007/s10208-012-9135-7

    Article  MathSciNet  MATH  Google Scholar 

  19. Candès, E., Recht, B.: Simple bounds for recovering low-complexity models. Math. Program. 141(1–2), 577–589 (2013). doi:10.1007/s10107-012-0540-0

  20. Fuchs, J.: On sparse representations in arbitrary redundant bases. IEEE Trans. Inf. Theory 50(6), 1341–1344 (2004). doi:10.1109/TIT.2004.828141

    Article  MathSciNet  MATH  Google Scholar 

  21. Dossal, C.: A necessary and sufficient condition for exact sparse recovery by \(\ell _1\) minimization. C. R. Acad. Sci. Paris 350(1–2), 117–120 (2012). doi:10.1016/j.crma.2011.12.014

    Article  MathSciNet  MATH  Google Scholar 

  22. Grasmair, M., Haltmeier, M., Scherzer, O.: Necessary and sufficient conditions for linear convergence of \(\ell _1\)-regularization. Commun. Pure Appl. Math. 64(2), 161–182 (2011). doi:10.1002/cpa.20350

    Article  MATH  Google Scholar 

  23. Candès, E., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006). doi:10.1109/TIT.2005.862083

    Article  MathSciNet  MATH  Google Scholar 

  24. Rockafellar, R.: Convex Analysis. No. 28 in Princeton Mathematics Ser. Princeton University Press, Princeton (1970)

    Google Scholar 

  25. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. No. 305–306 in Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1993)

    Book  Google Scholar 

  26. Borwein, J., Lewis, A.: Convex Analysis and Nonlinear Optimization—Theory and Examples. No. 3 in CMS Books in Mathematics. Springer, New York (2000)

    Book  Google Scholar 

  27. Polyak, B.: Sharp minima (1979). Presented at the IIASA Workshop on Generalized Lagrangians and Their Applications, IIASA, Laxenburg, Austria (1979)

  28. Polyak, B.: Introduction to Optimization. Optimization Software, New York (1987)

    MATH  Google Scholar 

  29. Burke, J., Ferris, M.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340–1359 (1993). doi:10.1137/0331063

    Article  MathSciNet  MATH  Google Scholar 

  30. Bertsekas, D.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  31. Bonnans, J., Gilbert, J., Lemaréchal, C., Sagastizábal, C.: Numerical Optimization—Theoretical and Practical Aspects, Universitext, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  32. Friedlander, M., Macêdo, I., Pong, T.: Gauge optimization and duality. SIAM J. Optim. 24(4), 1999–2022 (2014). doi:10.1137/130940785

    Article  MathSciNet  MATH  Google Scholar 

  33. Chvátal, V.: Linear Programming. W.H. Freeman, New York (1983)

    MATH  Google Scholar 

  34. Goldman, A., Tucker, A.: Theory of linear programming. In: Kuhn, H., Tucker, A. (eds.) Linear Inequalities and Related Systems, no. 38 in Annals of Mathematics Studies, pp. 53–97. Princeton University Press, Princeton (1956)

    Google Scholar 

  35. Balinski, M., Tucker, A.: Duality theory of linear programs, a constructive approach with applications. SIAM Rev. 11, 347–377 (1969). doi:10.1137/1011060

    Article  MathSciNet  MATH  Google Scholar 

  36. Saigal, R.: Linear Programming—A Modern Integrated Analysis. Kluwer Academic Publisher, Boston (1995)

    MATH  Google Scholar 

  37. Roos, C., Terlaky, T., Vial, J.P.: Theory and Algorithms for Linear Optimization—An Interior Point Approach. Wiley, Chichester (1997)

    MATH  Google Scholar 

  38. Wright, S.: Primal-Dual Interior-Point Methods. SIAM Publication, Philadelphia (1997)

    Book  MATH  Google Scholar 

  39. Armand, P., Gilbert, J., Jan-Jégou, S.: A feasible BFGS interior point algorithm for solving strongly convex minimization problems. SIAM J. Optim. 11, 199–222 (2000). doi:10.1137/S1052623498344720

    Article  MATH  Google Scholar 

  40. Gilbert, J., Gonzaga, C., Karas, E.: Examples of ill-behaved central paths in convex optimization. Math. Program. 103, 63–94 (2005). doi:10.1007/s10107-003-0460-0

    Article  MathSciNet  MATH  Google Scholar 

  41. Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bulletin de la Société Mathématique de France 93, 273–299 (1965). http://www.numdam.org/item?id=BSMF_1965__93__273_0

  42. Zhao, Y.B., Luo, Z.Q.: Constructing new weighted \(\ell _1\)-algorithms for the sparsest points of polyhedral sets. Tech. rep. (2016). http://www.optimization-online.org/DB_FILE/2016/08/5570.pdf

Download references

Author information

Authors and Affiliations

  1. INRIA Paris, 2 rue Simone Iff, CS 42112, 75589, Paris Cedex 12, France

    Jean Charles Gilbert

Authors
  1. Jean Charles Gilbert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean Charles Gilbert.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gilbert, J.C. On the Solution Uniqueness Characterization in the L1 Norm and Polyhedral Gauge Recovery. J Optim Theory Appl 172, 70–101 (2017). https://doi.org/10.1007/s10957-016-1004-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-016-1004-0

Keywords

Mathematics Subject Classification

Search

Navigation