Journal of Optimization Theory and Applications

, Volume 172, Issue 1, pp 70–101 | Cite as

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

  • Jean Charles Gilbert


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.


Basis pursuit Convex polyhedral function Gauge recovery L1 minimization Minkowski function Optimality conditions Sharp minimum Solution existence and uniqueness 

Mathematics Subject Classification

65K05 90C05 90C25 90C46 


  1. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, S., Donoho, D., Saunders, M.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20, 33–61 (1998). doi: 10.1137/S1064827596304010 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Natarajan, B.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24(2), 227–234 (2005). doi: 10.1137/S0097539792240406 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Candès, E., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(11), 4203–4215 (2005). doi: 10.1109/TIT.2005.858979 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Donoho, D.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). doi: 10.1109/TIT.2006.871582 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eldar, Y., Kutyniok, G. (eds.): Compressed Sensing: Theory and Applications. Cambridge University Press, Cambridge (2012)Google Scholar
  8. 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. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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 MathSciNetCrossRefGoogle Scholar
  11. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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 CrossRefzbMATHGoogle Scholar
  14. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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)Google Scholar
  16. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 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. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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 CrossRefzbMATHGoogle Scholar
  23. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rockafellar, R.: Convex Analysis. No. 28 in Princeton Mathematics Ser. Princeton University Press, Princeton (1970)Google Scholar
  25. 25.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. No. 305–306 in Grundlehren der mathematischen Wissenschaften. Springer, Berlin (1993)CrossRefGoogle Scholar
  26. 26.
    Borwein, J., Lewis, A.: Convex Analysis and Nonlinear Optimization—Theory and Examples. No. 3 in CMS Books in Mathematics. Springer, New York (2000)CrossRefGoogle Scholar
  27. 27.
    Polyak, B.: Sharp minima (1979). Presented at the IIASA Workshop on Generalized Lagrangians and Their Applications, IIASA, Laxenburg, Austria (1979)Google Scholar
  28. 28.
    Polyak, B.: Introduction to Optimization. Optimization Software, New York (1987)zbMATHGoogle Scholar
  29. 29.
    Burke, J., Ferris, M.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340–1359 (1993). doi: 10.1137/0331063 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Bertsekas, D.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)zbMATHGoogle Scholar
  31. 31.
    Bonnans, J., Gilbert, J., Lemaréchal, C., Sagastizábal, C.: Numerical Optimization—Theoretical and Practical Aspects, Universitext, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  32. 32.
    Friedlander, M., Macêdo, I., Pong, T.: Gauge optimization and duality. SIAM J. Optim. 24(4), 1999–2022 (2014). doi: 10.1137/130940785 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Chvátal, V.: Linear Programming. W.H. Freeman, New York (1983)zbMATHGoogle Scholar
  34. 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. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Saigal, R.: Linear Programming—A Modern Integrated Analysis. Kluwer Academic Publisher, Boston (1995)zbMATHGoogle Scholar
  37. 37.
    Roos, C., Terlaky, T., Vial, J.P.: Theory and Algorithms for Linear Optimization—An Interior Point Approach. Wiley, Chichester (1997)zbMATHGoogle Scholar
  38. 38.
    Wright, S.: Primal-Dual Interior-Point Methods. SIAM Publication, Philadelphia (1997)CrossRefzbMATHGoogle Scholar
  39. 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 CrossRefzbMATHGoogle Scholar
  40. 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 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Moreau, J.J.: Proximité et dualité dans un espace hilbertien. Bulletin de la Société Mathématique de France 93, 273–299 (1965).
  42. 42.
    Zhao, Y.B., Luo, Z.Q.: Constructing new weighted \(\ell _1\)-algorithms for the sparsest points of polyhedral sets. Tech. rep. (2016).

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.INRIA ParisParis Cedex 12France

Personalised recommendations