Perspective Reformulations of the CTA Problem with L2 Distances

Any institution that disseminates data in aggregated form has the duty to ensure that individual confidential information is not disclosed, either by not releasing data or by perturbing the released data while maintaining data utility. Controlled tabular adjustment CTA is a promising technique of the second type where a protected table that is close to the original one in some chosen distance is constructed. The choice of the specific distance shows a trade-off: although the Euclidean distance has been shown and is confirmed here to produce tables with greater “utility,” it gives rise to mixed integer quadratic problems MIQPs with pairs of linked semi-continuous variables that are more difficult to solve than the mixed integer linear problems corresponding to linear norms. We provide a novel analysis of perspective reformulations PRs for this special structure; in particular, we devise a projected PR P2R, which is piecewise-conic but simplifies to a nonseparable MIQP when the instance is symmetric. We then compare different formulations of the CTA problem, showing that the ones based on P2R most often obtain better computational results.

[1]  Claudio Gentile,et al.  Approximated perspective relaxations: a project and lift approach , 2016, Comput. Optim. Appl..

[2]  Juan José Salazar González.  Statistical confidentiality: Optimization techniques to protect tables , 2008, Comput. Oper. Res..

[3]  Sinan Gürel,et al.  A strong conic quadratic reformulation for machine-job assignment with controllable processing times , 2009, Oper. Res. Lett..

[4]  Matteo Fischetti,et al.  Solving the Cell Suppression Problem on Tabular Data with Linear Constraints , 2001, Manag. Sci..

[5]  Claudio Gentile,et al.  A computational comparison of reformulations of the perspective relaxation: SOCP vs. cutting planes , 2009, Oper. Res. Lett..

[6]  Jordi Castro.  Recent advances in optimization techniques for statistical tabular data protection , 2012, Eur. J. Oper. Res..

[7]  Oktay Günlük,et al.  Perspective Relaxation of Mixed Integer Nonlinear Programs with Indicator Variables , 2008, IPCO.

[8]  Nikolaos V. Sahinidis,et al.  Convex envelopes generated from finitely many compact convex sets , 2013, Math. Program..

[9]  James R. Luedtke,et al.  Some results on the strength of relaxations of multilinear functions , 2012, Math. Program..

[10]  Jordi Castro,et al.  A Shortest-Paths Heuristic for Statistical Data Protection in Positive Tables , 2007, INFORMS J. Comput..

[11]  Josep Domingo-Ferrer,et al.  Statistical Disclosure Control: Hundepool/Statistical Disclosure Control , 2012 .

[12]  Jordi Castro Pérez.  On assessing the disclosure risk of controlled adjustment methods for statistical tabular data , 2012 .

[13]  C. Gentile,et al.  Approximated Perspective Relaxations : a Project & Lift Approach , 2015 .

[14]  Nikolaos V. Sahinidis,et al.  Convex extensions and envelopes of lower semi-continuous functions , 2002, Math. Program..

[15]  Juan José SALAZAR-GONZÁLEZ.  Statistical confidentiality: Optimization techniques to protect tables , 2008 .

[16]  L. Willenborg,et al.  Elements of Statistical Disclosure Control , 2000 .

[17]  Jeff T. Linderoth,et al.  RELAXATION OF MINLPS WITH INDICATOR VARIABLES , 2007 .

[18]  Jordi Castro,et al.  Minimum-distance controlled perturbation methods for large-scale tabular data protection , 2006, Eur. J. Oper. Res..

[19]  Claudio Gentile,et al.  SDP diagonalizations and perspective cuts for a class of nonseparable MIQP , 2007, Oper. Res. Lett..

[20]  Nikolaos V. Sahinidis,et al.  Semidefinite Relaxations of Fractional Programs via Novel Convexification Techniques , 2001, J. Glob. Optim..

[21]  Oktay Günlük,et al.  Perspective Reformulation and Applications , 2012 .

[22]  Claudio Gentile,et al.  Projected Perspective Reformulations with Applications in Design Problems , 2011, Oper. Res..

[23]  Hassan L. Hijazi,et al.  Mixed Integer NonLinear Programs featuring "On/Off" constraints: convex analysis and applications , 2010, Electron. Notes Discret. Math..

[24]  C. Gentile,et al.  Tighter Approximated MILP Formulations for Unit Commitment Problems , 2009, IEEE Transactions on Power Systems.

[25]  Josep Domingo-Ferrer,et al.  A Critique of the Sensitivity Rules Usually Employed for Statistical Table Protection , 2002, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[26]  Josep Domingo-Ferrer,et al.  Statistical Disclosure Control , 2012 .

[27]  J. Pérez,et al.  Rounding methods for protecting EU-aggregates , 2007 .

[28]  Ignacio E. Grossmann,et al.  Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation , 2003, Comput. Optim. Appl..

[29]  M. Bacharach.  Matrix Rounding Problems , 1966 .

[30]  Doron Cohen.  Lecture Notes in Statistical Mechanics and Mesoscopics , 2011 .

[31]  Jean-Philippe P. Richard,et al.  KRANNERT GRADUATE SCHOOL OF MANAGEMENT , 2010 .

[32]  Sebastián Ceria,et al.  Convex programming for disjunctive convex optimization , 1999, Math. Program..

[33]  Jobst Heitzig,et al.  Federal Statistical Office Germany , 2005 .

[34]  Claudio Gentile,et al.  Perspective cuts for a class of convex 0–1 mixed integer programs , 2006, Math. Program..

[35]  Jordi Castro,et al.  Quadratic regularizations in an interior-point method for primal block-angular problems , 2011, Math. Program..

[36]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[37]  Claudio Gentile,et al.  Perspective cuts for 0-1 mixed integer programs , 2002 .

[38]  Jordi Castro,et al.  A heuristic block coordinate descent approach for controlled tabular adjustment , 2011, Comput. Oper. Res..

[39]  Sarah Giessing,et al.  Testing variants of minimum distance controlled tabular adjustment , 2008 .

[40]  James P. Kelly,et al.  Cell suppression: Disclosure protection for sensitive tabular data , 1992, Networks.