Mathematical Programming

, Volume 174, Issue 1–2, pp 77–97 | Cite as

Statistical inference of semidefinite programming

  • Alexander ShapiroEmail author
Full Length Paper Series B


In this paper we consider covariance structural models with which we associate semidefinite programming problems. We discuss statistical properties of estimates of the respective optimal value and optimal solutions when the ‘true’ covariance matrix is estimated by its sample counterpart. The analysis is based on perturbation theory of semidefinite programming. As an example we consider asymptotics of the so-called minimum trace factor analysis. We also discuss the minimum rank matrix completion problem and its SDP counterparts.


Semidefinite programming Minimum trace factor analysis Matrix completion problem Minimum rank Nondegeneracy Statistical inference Asymptotics 

Mathematics Subject Classification

62F12 62F30 90C22 



The author is indebted to anonymous referees for constructive comments which helped to improve the manuscript.


  1. 1.
    Alizadeh, F., Haeberly, J., Overton, M.: Complementarity and nondegeneracy in semidefinite programming. Math. Program. 77, 129–162 (1997)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arnold, V., Gusein-Zade, S., Varchenko, A.: Singularities of Differentiable Maps, vol. 1. Birkhäuser, Boston (1985)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bentler, P.: Lower-bound method for the dimension-free measurement of internal consistency. Soc. Sci. Res. 1, 343–357 (1972)CrossRefGoogle Scholar
  4. 4.
    Bonnans, J., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer Series in Operations Research. Springer, New York (2000)CrossRefzbMATHGoogle Scholar
  5. 5.
    Browne, M.: Generalized least squares estimators in the analysis of covariance structures. S. Afr. Stat. J. 8, 1–24 (1974)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Candés, E., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chandrasekaran, V., Sanghavi, S., Parilo, P., Willsky, A.: Rank-sparsity incoherence for matrix decomposition. SIAM J. Optim. 21, 572–596 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dür, M., Jargalsaikhan, B., Still, G.: Genericity results in linear conic programming—a tour d’horizon. Math. Oper. Res. 42, 77–94 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fazel, M.: Matrix rank minimization with applications. Ph.D. thesis, Stanford University (2002)Google Scholar
  10. 10.
    Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Springer, New York (1973)CrossRefzbMATHGoogle Scholar
  11. 11.
    Helmke, U., Moore, J.: Optimization and Dynamical Systems, 2nd edn. Springer, London (1996)zbMATHGoogle Scholar
  12. 12.
    Robinson, S.: Local structure of feasible sets in nonlinear programming II: mondegeneracy. Math. Program. Study 22, 217–230 (1984)CrossRefzbMATHGoogle Scholar
  13. 13.
    Scheinberg, K.: Handbook of semidefinite programming. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Parametric Linear Semidefinite Programming, Chap. 4, pp. 93–110. Kluwer Academic Publishers, Boston (2000)Google Scholar
  14. 14.
    Shapiro, A.: Rank reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis. Psychometrika 47, 187–199 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shapiro, A.: Weighted minimum trace factor analysis. Psychometrika 47, 243–264 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shapiro, A.: Extremal problems on the set of nonnegative definite matrices. Linear Algebra Appl. 67, 7–18 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Math. Program. 77, 301–320 (1997)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Shapiro, A.: Duality, optimality conditions, and perturbation analysis. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming, Chap. 4, pp. 67–92. Kluwer Academic Publishers, Boston (2000)CrossRefGoogle Scholar
  19. 19.
    Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming: Modeling and Theory, 2nd edn. SIAM, Philadelphia (2014)zbMATHGoogle Scholar
  20. 20.
    Shapiro, A., Fan, M.: On eigenvalue optimization. SIAM J. Optim. 5, 552–569 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shapiro, A., Ten Berge, J.: Statistical inference of minimum rank factor analysis. Psychometrika 67, 79–94 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ten Berge, J., Snijders, T., Zegers, F.: Computational aspects of the greatest lower bound to the reliability and constrained minimum trace factor analysis. Psychometrika 46, 201–213 (1981)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

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