Statistical inference of semidefinite programming with multiple parameters

Jiani Wang; Liwei Zhang

doi:10.3934/jimo.2019015

Article Contents

2020, Volume 16, Issue 3: 1527-1538. Doi: 10.3934/jimo.2019015

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Statistical inference of semidefinite programming with multiple parameters

Jiani Wang^, and
Liwei Zhang

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

^* Corresponding author: Jiani Wang
^* Corresponding author: Jiani Wang

Received: April 30, 2018

Revised: September 30, 2018

Early access: March 2019

Published: March 2020

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

Abstract Full Text(HTML) Figure(0) / Table(3) Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: Primary: 90C22, 90C31, 90C46; Secondary: 62D05, 62F12.

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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

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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

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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
100	$ x_2 $	-0.005239017	0.2040734	0.2006396	0.951
400	$ x_1 $	0.003114901	0.09937129	0.1003198	0.948
400	$ x_2 $	0.004845715	0.1005173	0.1003198	0.946
1000	$ x_1 $	-0.0001884376	0.06216153	0.06344781	0.943
1000	$ x_2 $	0.005075925	0.06360439	0.06344781	0.952

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