May  2020, 16(3): 1527-1538. doi: 10.3934/jimo.2019015

Statistical inference of semidefinite programming with multiple parameters

School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

* Corresponding author: Jiani Wang

Received  May 2018 Revised  October 2018 Published  May 2020 Early access  March 2019

Fund Project: Jiani Wang is supported by NNSFC grant Nos. 11571059, 11731013 and 91330206

The parameters in the semidefinite programming problems generated by the average of a sample, may lead to the deviation of the optimal value and optimal solutions due to the uncertainty of the data. The statistical properties of estimates of the optimal value and the optimal solutions are given in this paper, when the estimated parameters are both in the objective function and in the constraints. This analysis is mainly based on the theory of the linear programming and the perturbation theory of the semidefinite programming.

Citation: Jiani Wang, Liwei Zhang. Statistical inference of semidefinite programming with multiple parameters. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1527-1538. doi: 10.3934/jimo.2019015
References:
[1]

F. AlizadehJ. P. A. Haeberly and M. L. Overton, Complementarity and nondegeneracy in semidefinite programming, Mathematical Programming, 77 (1997), 111-128.  doi: 10.1007/BF02614432.

[2]

H. Bauer, Measure and Integration Theory (Vol. 26), Walter de Gruyter, Berlin, 2001. doi: 10.1515/9783110866209.

[3]

P. Billingsley, Probability and Measure (3rd ed.), Wiley series in probability and mathematical statistics, New York, 1995.

[4]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.

[5]

M. W. Browne, Fitting the factor analysis model, ETS Research Report Series, 1967 (1967), i–43.

[6]

E. J. Candés and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational mathematics, 9 (2009), 717-772.  doi: 10.1007/s10208-009-9045-5.

[7]

M. DürB. Jargalsaikhan and G. Still, Genericity results in linear conic programming–a tour d'horizon, Mathematics of Operations Research, 42 (2017), 77-94.  doi: 10.1287/moor.2016.0793.

[8]

H. Fischer, A History of the Central Limit Theorem: From Classical to Modern Probability Theory, Springer, New York, 2011. doi: 10.1007/978-0-387-87857-7.

[9]

A. Hald, A History of Mathematical Statistics from 1750 to 1930, Wiley, 1998.

[10]

J.-B. Hiriart-Urruty, Fundamentals of Convex Analysis, Springer-Verlag, New York, 2001. doi: 10.1007/978-3-642-56468-0.

[11]

H. B. Mann and A. Wald, On stochastic limit and order relationships, Annals of Mathematical Statistics, 14 (1943), 217-226.  doi: 10.1214/aoms/1177731415.

[12]

A. Shapiro, Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis, Psychometrika, 47 (1982), 187-199.  doi: 10.1007/BF02296274.

[13]

A. Shapiro, Statistical inference of semidefinite programming, Mathematical Programming, (2018), 1–21, Available from: http://www.optimization-online.org/DB\_HTML/2017/01/5842.html. doi: 10.1007/s10107-018-1250-z.

[14]

A. Shapiro and K. Scheinberg, Duality and optimality conditions, in Handbook of Semidefinite Programming, Springer, Boston, MA, 27 (2000), 66–110. doi: 10.1007/978-1-4615-4381-7_4.

[15]

A. Shapiro and J. M. F. Ten Berge, Statistical inference of minimum rank factor analysis, Psychometrika, 67 (2002), 79-94.  doi: 10.1007/BF02294710.

[16]

E. Slutsky, Uber stochastische asymptoten und grenzwerte, Metron, 5 (1925), 3-89. 

[17]

M. J. Todd, Semidefinite optimization, Acta Numerica, 10 (2001), 515-560.  doi: 10.1017/S0962492901000071.

[18] A. W. Van der Vaart, Asymptotic Statistics, Cambridge University Press, New York, 1998.  doi: 10.1017/CBO9780511802256.

show all references

References:
[1]

F. AlizadehJ. P. A. Haeberly and M. L. Overton, Complementarity and nondegeneracy in semidefinite programming, Mathematical Programming, 77 (1997), 111-128.  doi: 10.1007/BF02614432.

[2]

H. Bauer, Measure and Integration Theory (Vol. 26), Walter de Gruyter, Berlin, 2001. doi: 10.1515/9783110866209.

[3]

P. Billingsley, Probability and Measure (3rd ed.), Wiley series in probability and mathematical statistics, New York, 1995.

[4]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.

[5]

M. W. Browne, Fitting the factor analysis model, ETS Research Report Series, 1967 (1967), i–43.

[6]

E. J. Candés and B. Recht, Exact matrix completion via convex optimization, Foundations of Computational mathematics, 9 (2009), 717-772.  doi: 10.1007/s10208-009-9045-5.

[7]

M. DürB. Jargalsaikhan and G. Still, Genericity results in linear conic programming–a tour d'horizon, Mathematics of Operations Research, 42 (2017), 77-94.  doi: 10.1287/moor.2016.0793.

[8]

H. Fischer, A History of the Central Limit Theorem: From Classical to Modern Probability Theory, Springer, New York, 2011. doi: 10.1007/978-0-387-87857-7.

[9]

A. Hald, A History of Mathematical Statistics from 1750 to 1930, Wiley, 1998.

[10]

J.-B. Hiriart-Urruty, Fundamentals of Convex Analysis, Springer-Verlag, New York, 2001. doi: 10.1007/978-3-642-56468-0.

[11]

H. B. Mann and A. Wald, On stochastic limit and order relationships, Annals of Mathematical Statistics, 14 (1943), 217-226.  doi: 10.1214/aoms/1177731415.

[12]

A. Shapiro, Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis, Psychometrika, 47 (1982), 187-199.  doi: 10.1007/BF02296274.

[13]

A. Shapiro, Statistical inference of semidefinite programming, Mathematical Programming, (2018), 1–21, Available from: http://www.optimization-online.org/DB\_HTML/2017/01/5842.html. doi: 10.1007/s10107-018-1250-z.

[14]

A. Shapiro and K. Scheinberg, Duality and optimality conditions, in Handbook of Semidefinite Programming, Springer, Boston, MA, 27 (2000), 66–110. doi: 10.1007/978-1-4615-4381-7_4.

[15]

A. Shapiro and J. M. F. Ten Berge, Statistical inference of minimum rank factor analysis, Psychometrika, 67 (2002), 79-94.  doi: 10.1007/BF02294710.

[16]

E. Slutsky, Uber stochastische asymptoten und grenzwerte, Metron, 5 (1925), 3-89. 

[17]

M. J. Todd, Semidefinite optimization, Acta Numerica, 10 (2001), 515-560.  doi: 10.1017/S0962492901000071.

[18] A. W. Van der Vaart, Asymptotic Statistics, Cambridge University Press, New York, 1998.  doi: 10.1017/CBO9780511802256.
Table 1.  $ \hat{\vartheta}_N $ in the case that the optimal solution is not unique
N Bias SD SE CP
100 -0.01575607 0.09802304 0.1026943 0.959
300 -0.008588234 0.05875263 0.05928791 0.947
800 -0.005730269 0.03494695 0.03630683 0.953
N Bias SD SE CP
100 -0.01575607 0.09802304 0.1026943 0.959
300 -0.008588234 0.05875263 0.05928791 0.947
800 -0.005730269 0.03494695 0.03630683 0.953
Table 2.  $ \hat{\vartheta}_N $ with the unique optimal solution
N Bias SD SE CP
100 -0.008575782 0.2752905 0.283196 0.954
300 0.000433069 0.1598366 0.1635033 0.953
800 0.002228357 0.1022441 0.1001249 0.948
N Bias SD SE CP
100 -0.008575782 0.2752905 0.283196 0.954
300 0.000433069 0.1598366 0.1635033 0.953
800 0.002228357 0.1022441 0.1001249 0.948
Table 3.  $ \hat{x}_N $ with the unique optimal solution
N x Bias SD SE CP
100 $ x_1 $ 0.001119686 0.1960365 0.2006396 0.956
$ x_2 $ -0.005239017 0.2040734 0.2006396 0.951
400 $ x_1 $ 0.003114901 0.09937129 0.1003198 0.948
$ x_2 $ 0.004845715 0.1005173 0.1003198 0.946
1000 $ x_1 $ -0.0001884376 0.06216153 0.06344781 0.943
$ x_2 $ 0.005075925 0.06360439 0.06344781 0.952
N x Bias SD SE CP
100 $ x_1 $ 0.001119686 0.1960365 0.2006396 0.956
$ x_2 $ -0.005239017 0.2040734 0.2006396 0.951
400 $ x_1 $ 0.003114901 0.09937129 0.1003198 0.948
$ x_2 $ 0.004845715 0.1005173 0.1003198 0.946
1000 $ x_1 $ -0.0001884376 0.06216153 0.06344781 0.943
$ x_2 $ 0.005075925 0.06360439 0.06344781 0.952
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