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January  2016, 12(1): 317-336. doi: 10.3934/jimo.2016.12.317

An alternating direction method for solving a class of inverse semi-definite quadratic programming problems

Yue Lu 1, , Ying-En Ge 2, and  Li-Wei Zhang 1,

1. 

Institute of Operations Research and Control Theory, School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China, China
2. 

School of Transportation and Logistics, Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China

Received  April 2013 Revised  January 2015 Published  April 2015

In this paper, we propose an alternating-direction-type numerical method to solve a class of inverse semi-definite quadratic programming problems. An explicit solution to one direction subproblem is given and the other direction subproblem is proved to be a convex quadratic programming problem over positive semi-definite symmetric matrix cone. We design a spectral projected gradient method for solving the quadratic matrix optimization problem and demonstrate its convergence. Numerical experiments illustrate that our method can solve inverse semi-definite quadratic programming problems efficiently.
Keywords: Inverse semi-definite programming, spectral projected gradient method, BB-step, alternating direction method, convex semi-definite quadratic programming..
Mathematics Subject Classification: Primary: 90C20, 90C22; Secondary: 49M2.
Citation: Yue Lu, Ying-En Ge, Li-Wei Zhang. An alternating direction method for solving a class of inverse semi-definite quadratic programming problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 317-336. doi: 10.3934/jimo.2016.12.317
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SIAM Journal on Scientific Computing, 33 (2011), 250-278. doi: 10.1137/090777761.  Google Scholar

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Journal of Computational and Applied Mathematics, 72 (1996), 261-273. doi: 10.1016/0377-0427(95)00277-4.  Google Scholar

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Journal of Computational and Applied Mathematics, 106 (1999), 345-359. doi: 10.1016/S0377-0427(99)00080-1.  Google Scholar

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show all references

References:
[1]

IEEE Transactions on image processing, 19 (2010), 2345-2356. doi: 10.1109/TIP.2010.2047910.  Google Scholar

[2]

Operations Research, 49 (2001), 771-783. doi: 10.1287/opre.49.5.771.10607.  Google Scholar

[3]

Networks, 40 (2002), 181-187. doi: 10.1002/net.10048.  Google Scholar

[4]

IMA Journal of Numerical Analysis, 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141.  Google Scholar

[5]

IEEE Transactions on Automatic Control, 21 (1976), 174-184. doi: 10.1109/TAC.1976.1101194.  Google Scholar

[6]

SIAM Journal on Optimization, 10 (2000), 1196-1211. doi: 10.1137/S1052623497330963.  Google Scholar

[7]

E. Birgin, J. Martínez and M. Raydan, Spectral projected gradient methods: reviews and perspective,, Available from: , ().   Google Scholar

[8]

Foundations and Trends in Machine Learning, 3 (2011), 1-122. doi: 10.1561/2200000016.  Google Scholar

[9]

Mathematical Programming, 53 (1992), 45-61. doi: 10.1007/BF01585693.  Google Scholar

[10]

Journal of Global Optimization, 15 (1999), 213-218. doi: 10.1023/A:1008360312607.  Google Scholar

[11]

IMA Journal on Numerical Analysis, 22 (2002), 1-10. doi: 10.1093/imanum/22.1.1.  Google Scholar

[12]

Transactions of the American Mathematical Society, 82 (1956), 421-439. doi: 10.1090/S0002-9947-1956-0084194-4.  Google Scholar

[13]

SIAM Journal on Numerical Analysis, 36 (1999), 275-289. doi: 10.1137/S003614299427315X.  Google Scholar

[14]

Computers & Mathematics with Applications, 2 (1976), 17-40. doi: 10.1016/0898-1221(76)90003-1.  Google Scholar

[15]

SIAM, Philadelphia, 1989. doi: 10.1137/1.9781611970838.  Google Scholar

[16]

SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.  Google Scholar

[17]

Journal of Combinatorial Optimization, 8 (2004), 329-361. doi: 10.1023/B:JOCO.0000038914.26975.9b.  Google Scholar

[18]

Mathematical Programming, 92 (2002), 103-118. doi: 10.1007/s101070100280.  Google Scholar

[19]

SIAM Journal on Optimization, 22 (2012), 313-340. doi: 10.1137/110822347.  Google Scholar

[20]

Operations Research Letters, 33 (2005), 319-330. doi: 10.1016/j.orl.2004.04.007.  Google Scholar

[21]

Journal of the Society for Industrial and Applied Mathematics, 3 (1955), 28-41. doi: 10.1137/0103003.  Google Scholar

[22]

IMA Journal of Numerical Analysis, 13 (1993), 321-326. doi: 10.1093/imanum/13.3.321.  Google Scholar

[23]

SIAM Journal on Optimization, 7 (1997), 26-33. doi: 10.1137/S1052623494266365.  Google Scholar

[24]

Springer-Verlag, New York, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[25]

Mathematical Programming, 114 (2008), 349-391. doi: 10.1007/s10107-007-0105-9.  Google Scholar

[26]

SIAM Journal on Control and Optimization, 29 (1991), 119-138. doi: 10.1137/0329006.  Google Scholar

[27]

Mathematical Programming, 48 (1990), 249-263. doi: 10.1007/BF01582258.  Google Scholar

[28]

Journal of Computational and Applied Mathematics, 223 (2009), 485-498. doi: 10.1016/j.cam.2008.01.028.  Google Scholar

[29]

Journal of Industrial and Management Optimization, 5 (2009), 319-339. doi: 10.3934/jimo.2009.5.319.  Google Scholar

[30]

SIAM Journal on Scientific Computing, 33 (2011), 250-278. doi: 10.1137/090777761.  Google Scholar

[31]

Journal of Computational and Applied Mathematics, 72 (1996), 261-273. doi: 10.1016/0377-0427(95)00277-4.  Google Scholar

[32]

Journal of Computational and Applied Mathematics, 106 (1999), 345-359. doi: 10.1016/S0377-0427(99)00080-1.  Google Scholar

[33]

European Journal of Operations Research, 124 (2000), 77-88. doi: 10.1016/S0377-2217(99)00122-8.  Google Scholar

[34]

Journal of Combinatorial Optimization, 3 (1999), 127-139. doi: 10.1023/A:1009829525096.  Google Scholar

[35]

Applied Mathematics and Optimization, 61 (2010), 57-83. doi: 10.1007/s00245-009-9075-z.  Google Scholar

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