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  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 & 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

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

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18] A. W. Van der Vaart, Asymptotic Statistics, Cambridge University Press, New York, 1998.  doi: 10.1017/CBO9780511802256.  Google Scholar
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
[1]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[2]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHum approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[3]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[4]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[5]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[6]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054

[7]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[8]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[9]

Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044

[10]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[11]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[12]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[13]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[14]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[15]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (89)
  • HTML views (647)
  • Cited by (0)

Other articles
by authors

[Back to Top]