Skip to main content
Springer Nature Link
Log in

A Second Order Cone Formulation of Continuous CTA Model

  • Conference paper
  • First Online:
Privacy in Statistical Databases (PSD 2016)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 9867))

Included in the following conference series:

Abstract

In this paper we consider a minimum distance Controlled Tabular Adjustment (CTA) model for statistical disclosure limitation (control) of tabular data. The goal of the CTA model is to find the closest safe table to some original tabular data set that contains sensitive information. The measure of closeness is usually measured using \(\ell _1\) or \(\ell _2\) norm; with each measure having its advantages and disadvantages. Recently, in [4] a regularization of the \(\ell _1\)-CTA using Pseudo-Huber function was introduced in an attempt to combine positive characteristics of both \(\ell _1\)-CTA and \(\ell _2\)-CTA. All three models can be solved using appropriate versions of Interior-Point Methods (IPM). It is known that IPM in general works better on well structured problems such as conic optimization problems, thus, reformulation of these CTA models as conic optimization problem may be advantageous. We present reformulation of Pseudo-Huber-CTA, and \(\ell _1\)-CTA as Second-Order Cone (SOC) optimization problems and test the validity of the approach on the small example of two-dimensional tabular data set.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Andersen, E.D.: MOSEK solver (2016). https://mosek.com/resources/doc

  2. Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program. 95(1), 3–51 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andersen, E.D., Roos, C., Terlaky, T.: On implementing a primal-dual interior-point method for conic quadratic optimization. Math. Program. 95(2), 249–277 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Castro, J.: A CTA model based on the huber function. In: Domingo-Ferrer, J. (ed.) PSD 2014. LNCS, vol. 8744, pp. 79–88. Springer, Heidelberg (2014)

    Google Scholar 

  5. Castro, J.: An interior-point approach for primal block-angular problems. Comput. Optim. Appl. 36, 195–219 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Castro, J.: On assessing the disclosure risk of controlled adjustment methods for statistical tabular data. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 20, 921–941 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Castro, J.: Minimum-distance controlled perturbation methods for large-scale tabular data protection. Eur. J. Oper. Res. 171, 39–52 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Castro, J.: Recent advances in optimization techniques for statistical tabular data protection. Eur. J. Oper. Res. 216, 257–269 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Castro, J., Cuesta, J.: Quadratic regularization in an interior-point method for primal block-angular problems. Math. Program. 130, 415–445 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Castro, J., Cuesta, J.: Solving \(\ell _1\)-CTA in 3D tables by an interior-point method for primal block-angular problems. TOP 21, 25–47 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Castro, J., González, J.A.: Assessing the information loss of controlled adjustment methods in two-way tables. In: Domingo-Ferrer, J. (ed.) PSD 2014. LNCS, vol. 8744, pp. 11–23. Springer, Heidelberg (2014)

    Google Scholar 

  12. Castro, J., Gonzalez, J.A.: A fast CTA method without complicating binary decisions. In: Documents of the Joint UNECE/Eurostat Work Session on Statistical Data Confidentiality, Statistics Canada, Ottawa, pp. 1–7 (2013)

    Google Scholar 

  13. Castro, J., Gonzalez, J.A.: A multiobjective LP approach for controlled tabular adjustment in statistical disclosure control. Working paper, Department of Statistics and Operations Research, Universitat Politecnica de Catalunya (2014)

    Google Scholar 

  14. Castro, J., Giessing, S.: Testing variants of minimum distance controlled tabular adjustment. In: Monographs of Official Statistics, Eurostat-Office for Official Publications of the European Communities, Luxembourg, pp. 333–343 (2006)

    Google Scholar 

  15. Dandekar, R.A., Cox, L.H.: Synthetic tabular data: an alternative to complementary cell suppression. Manuscript, Energy Information Administration, U.S. (2002)

    Google Scholar 

  16. Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. Oxford University Press, New York (1994)

    MATH  Google Scholar 

  17. Fountoulakis, K., Gondzio, J.: A second-order method for strongly convex L1-regularization problems. Technical report ERGO-14-005, School of Mathematics, The University of Edinburgh (2014)

    Google Scholar 

  18. Gu, G.: Interior-point methods for symmetric optimization. Ph.d. thesis, TU Delft (2009)

    Google Scholar 

  19. Hundepool, A., Domingo-Ferrer, J., Franconi, L., Giessing, S., Schulte Nordholt, E., Spicer, K., De Wolf, P.-P.: Statistical Disclosure Control. Wiley, Chichester (2012)

    Book  Google Scholar 

  20. Karmarkar, N.: A polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  21. Karr, A.F., Kohnen, C.N., Oganian, A., Reiter, J.P., Sanil, A.P.: A framework for evaluating the utility of data altered to protect confidentiality. Am. Stat. 60(3), 224–232 (2006)

    Article  MathSciNet  Google Scholar 

  22. Lesaja, G.: Introducing interior-point methods for introductory operations research courses and/or linear programming courses. Open Oper. Res. J. 3, 1–12 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lesaja, G., Slaughter, V.: Interior-point algorithms for a class of convex optimization problems. Yugoslav J. Oper. Res. 19(3), 239–248 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lesaja, G., Roos, C.: Kernel-based interior-point methods for monotone linear complementarity problems over symmetric cones. J. Optim. Theor. Appl. 150(3), 444–474 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ben-Tal, A., Nemirovski, A.: Lectures in Modern Convex Optimization: Analysis, Algorithms and Engineering Applications. MPS/SIAM Series in Optimization. SIAM, Philadelphia (2001)

    Book  MATH  Google Scholar 

  26. Nesterov, Y., Nemirovski, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics, vol. 13. SIAM, Philadelphia (1994)

    Book  Google Scholar 

  27. Oganian, A.: Security and information loss in statistical database protection. Ph.d. thesis, Universitat Politecnica de Catalunya (2003)

    Google Scholar 

  28. Roos, C., Terlaky, T., Vial, J.P.: Theory and Algorithms for Linear Optimization. An Interior-Point Approach. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  29. Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1996)

    MATH  Google Scholar 

Download references

Acknowledgments

The first author would like to thank Erling Andersen and Florian Jarre for the constructive discussion regarding the SOC model and Iryna Petrenko for her help in performing the calculations described in Sect. 5.

The authors would like to express their appreciation to Donald Malec for his careful reading of the paper and many useful suggestions.

Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors only and do not necessarily reflect the views of the Centers for Disease Control and Prevention.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Georgia Southern University, P.O. Box 8093, Statesboro, GA, 30460-8093, USA

    Goran Lesaja & Anna Oganian

  2. Department of Statistics and Operations Research, Universitat Politècnica de Catalunya, Jordi Girona 1–3, 08034, Barcelona, Catalonia, Spain

    Jordi Castro

  3. National Center for Health Statistics, 3311 Toledo Road, Hyatsville, MD, 20782, USA

    Anna Oganian

Authors
  1. Goran Lesaja
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jordi Castro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anna Oganian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Goran Lesaja .

Editor information

Editors and Affiliations

  1. Universitat Rovira i Virgili, Tarragona, Spain

    Josep Domingo-Ferrer

  2. University of Zagreb, Zagreb, Croatia

    Mirjana Pejić-Bach

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Lesaja, G., Castro, J., Oganian, A. (2016). A Second Order Cone Formulation of Continuous CTA Model. In: Domingo-Ferrer, J., Pejić-Bach, M. (eds) Privacy in Statistical Databases. PSD 2016. Lecture Notes in Computer Science(), vol 9867. Springer, Cham. https://doi.org/10.1007/978-3-319-45381-1_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45381-1_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45380-4

  • Online ISBN: 978-3-319-45381-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics