zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A note on the complexity of L p minimization. (English) Zbl 1226.90076
Summary: We discuss the L p (0p<1) minimization problem arising from sparse solution construction and compressed sensing. For any fixed 0<p<1, we prove that finding the global minimal value of the problem is strongly NP-Hard, but computing a local minimizer of the problem can be done in polynomial time. We also develop an interior-point potential reduction algorithm with a provable complexity bound and demonstrate preliminary computational results of effectiveness of the algorithm.
90C26Nonconvex programming, global optimization
90C51Interior-point methods
[1]Berinde, R., Gilbert, A.C., Indyk, P., Karloff, H.J., Strauss, M.J.: Combining geometry and combinatorics: a unified approach to sparse signal recovery, preprint (2008)
[2]Bertsekas D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)
[3]Bertsimas D., Tsitsiklis J.: Introduction to Linear Optimization, pp. 414. Athena Scientific, Belmont (1997)
[4]Blumensath T., Davies M.: Iterative hard thresholding for compressed sensing. Appl. Comp. Harmonic Anal. 27, 265–274 (2009) · Zbl 1174.94008 · doi:10.1016/j.acha.2009.04.002
[5]Borwein, J.M., Luke, D.R.: Entropic Regularization of the ł0 function, In: Fixed-point algorithms for Inverse Problems in Science and Engineering in Springer optimization and its Applications (2010)
[6]Bruckstein A.M., Donoho D.L., Elad M.: From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1), 34C81 (2009)
[7]Candès E.J., Tao T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51, 4203–4215 (2005) · Zbl 1264.94121 · doi:10.1109/TIT.2005.858979
[8]Candès E.J., Tao T.: Near optimal signal recovery from random projections: Universal encoding strategies. IEEE Trans. Inf. Theory 52, 5406–5425 (2006) · Zbl 05455295 · doi:10.1109/TIT.2006.885507
[9]Chartrand, R.: Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process. Lett. 14–10 (2009)
[10]Chartrand, R.: Fast algorithms for nonconvex compressive sensing: MRI reconstruction from very few data, In: IEEE International Symposium on Biomedical Imaging (ISBI) (2009)
[11]Chen,X., Xu, F., Ye, Y.: Lower bound theory of nonzero entries in solutions of 2- p minimization, Technical Report, The Hong Kong, Polytechnic University (2009)
[12]Donoho, D.: For most large underdetermined systems of linear equations the minimal l 1-norm solution is also the sparsest Solution, Technical Report, Stanford University (2004)
[13]Fan J., Li R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Soc. 96, 1348–1360 (2001) · Zbl 1073.62547 · doi:10.1198/016214501753382273
[14]Gilbert, A.C., Muthukrishnan, M., Strauss, M.J.: Approximation of functions over redundant dictionaries using coherence, In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2003)
[15]Garey M.R., Johnson D.S.: Strong NP-Completeness: results, motivation examples and implications. J. Assoc. Comput. Mach. 25, 499–508 (1978) · Zbl 0379.68035 · doi:10.1145/322077.322090
[16]Garey M.R., Johnson D.S.: Computers and Intractability A Guide to the Theory of NP-Completeness, pp. 96–105. W. H. Freeman, San Francisco (1979)
[17]Mourad N., Reilly P.: Minimizing nonconvex functions for sparse vector reconstruction. IEEE Trans. Signal Process. 58, 3485–3496 (2010) · doi:10.1109/TSP.2010.2046900
[18]Nesterov Y., Nemirovski A.: Interior Point Polynomial Methods in Convex Programming. SIAM, Philadelphia, PA (1994)
[19]Natarajan B.K.: Sparse approximate solutions to linear systems. SIAM J. Comput. 24, 227–234 (1995) · Zbl 0827.68054 · doi:10.1137/S0097539792240406
[20]Terlaky T.: On p programming. Eur. J. Oper. Res. 22, 70–100 (1985) · Zbl 0577.90062 · doi:10.1016/0377-2217(85)90116-X
[21]Tropp J.A.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Info. Theory 50, 2231–2242 (2004) · Zbl 05455041 · doi:10.1109/TIT.2004.834793
[22]Tropp, J., Wright, S.J.: Computational methods for sparse solutions of linear inverse problems. In: Proceedings of the IEEE (2010)
[23]Vavasis S.A.: Polynomial time weak approximation algorithms for quadratic programming. In: Floudas, C.A., Pardalos, P.M. (eds) Complexity in Numerical Optimization, World Scientific, New Jersey (1993)
[24]Vazirani V.: Approximation Algorithms. Springer, Berlin (2003)
[25]Xue G., Ye Y.: An efficient algorithm for minimizing a sum of P-norms. SIAM J. Optim. 10, 551–579 (2000) · Zbl 0955.68126 · doi:10.1137/S1052623497327088
[26]Ye Y.: On the complexity of approximating a KKT point of quadratic programming. Math. Progr. 80, 195–211 (1998) · Zbl 0894.90117 · doi:10.1007/BF01581726
[27]Ye Y.: Interior Point Algorithms: Theory and Analysis. Wiley, New York (1997)