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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

 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.
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 and Management Optimization, 2016, 12 (1) : 317-336. doi: 10.3934/jimo.2016.12.317
##### References:
 [1] M. Afonso, J. Bioucas-Dias and M. Figueiredo, Fast image recovery using variable splitting and constrained optimization, IEEE Transactions on image processing, 19 (2010), 2345-2356. doi: 10.1109/TIP.2010.2047910. [2] R. Ahuja and J. Orlin, Inverse optimization, Operations Research, 49 (2001), 771-783. doi: 10.1287/opre.49.5.771.10607. [3] R. Ahuja and J. Orlin, Combinatorial algorithms for inverse network flow problems, Networks, 40 (2002), 181-187. doi: 10.1002/net.10048. [4] J. Barzilai and J. M. Borwein, Two point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141. [5] D. Bertsekas, On the Goldstein-Levitin-Polyak gradient projection method, IEEE Transactions on Automatic Control, 21 (1976), 174-184. doi: 10.1109/TAC.1976.1101194. [6] E. Birgin, J. Martínez and M. Raydan, Nonmonotone spectral projected gradient methods on convex sets, SIAM Journal on Optimization, 10 (2000), 1196-1211. doi: 10.1137/S1052623497330963. [7] E. Birgin, J. Martínez and M. Raydan, Spectral projected gradient methods: reviews and perspective,, Available from: , (). [8] S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2011), 1-122. doi: 10.1561/2200000016. [9] W. Burton and P. Toint, On an instance of the inverse shortest paths problem, Mathematical Programming, 53 (1992), 45-61. doi: 10.1007/BF01585693. [10] M. Cai, X. Yang and J. Zhang, The complexity analysis of the inverse center location problem, Journal of Global Optimization, 15 (1999), 213-218. doi: 10.1023/A:1008360312607. [11] Y. Dai and L. Liao, R-linear convergence of the Barzilai and Borwein gradient method, IMA Journal on Numerical Analysis, 22 (2002), 1-10. doi: 10.1093/imanum/22.1.1. [12] J. Douglas and H. Rachford, On the numerical solution of the heat conduction problem in two and three space variables, Transactions of the American Mathematical Society, 82 (1956), 421-439. doi: 10.1090/S0002-9947-1956-0084194-4. [13] A. Friedlander, J. M. Martínez, B. Molina and M. Raydan, Gradient method with retards and generalizations, SIAM Journal on Numerical Analysis, 36 (1999), 275-289. doi: 10.1137/S003614299427315X. [14] D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite-element approximations, Computers & Mathematics with Applications, 2 (1976), 17-40. doi: 10.1016/0898-1221(76)90003-1. [15] R. Glowinski and P. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989. doi: 10.1137/1.9781611970838. [16] T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [17] C. Heuberger, Inverse combinatorial optimization: A survey on problems, methods and results, Journal of Combinatorial Optimization, 8 (2004), 329-361. doi: 10.1023/B:JOCO.0000038914.26975.9b. [18] B. He, L. Liao, D. Han and H. Yang, A new inexact alternating direction method for monotone variational inequalities, Mathematical Programming, 92 (2002), 103-118. doi: 10.1007/s101070100280. [19] B. He, M. Tao and X. Yuan, Alternating direction method with Gaussian back substitution for separable convex programming, SIAM Journal on Optimization, 22 (2012), 313-340. doi: 10.1137/110822347. [20] G. Iyengar and W. Kang, Inverse conic programming and applications, Operations Research Letters, 33 (2005), 319-330. doi: 10.1016/j.orl.2004.04.007. [21] D. Peaceman and H. Rachford, The numerical solution of parabolic elliptic differential equations, Journal of the Society for Industrial and Applied Mathematics, 3 (1955), 28-41. doi: 10.1137/0103003. [22] M. Raydan, On the Barzilai and Borwein choice of steplength for the gradient method, IMA Journal of Numerical Analysis, 13 (1993), 321-326. doi: 10.1093/imanum/13.3.321. [23] M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM Journal on Optimization, 7 (1997), 26-33. doi: 10.1137/S1052623494266365. [24] R. Rockafellar and R. Wets, Variational Analysis, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-642-02431-3. [25] D. Sun, J. Sun and L. Zhang, The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming, Mathematical Programming, 114 (2008), 349-391. doi: 10.1007/s10107-007-0105-9. [26] P. Tseng, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM Journal on Control and Optimization, 29 (1991), 119-138. doi: 10.1137/0329006. [27] P. Tseng, Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming, Mathematical Programming, 48 (1990), 249-263. doi: 10.1007/BF01582258. [28] X. Xiao, L. Zhang and J. Zhang, A smoothing Newton method for a type of inverse semi-definite quadratic programming problem, Journal of Computational and Applied Mathematics, 223 (2009), 485-498. doi: 10.1016/j.cam.2008.01.028. [29] X. Xiao, L. Zhang and J. Zhang, On convergence of augmented Lagrange method for inverse semi-definite quadratic programming problems, Journal of Industrial and Management Optimization, 5 (2009), 319-339. doi: 10.3934/jimo.2009.5.319. [30] J. Yang and Y. Zhang, Alternating direction algorithms for $l_1$-problems in compressive sensing, SIAM Journal on Scientific Computing, 33 (2011), 250-278. doi: 10.1137/090777761. [31] J. Zhang and Z. Liu, Calculating some inverse linear programming problems, Journal of Computational and Applied Mathematics, 72 (1996), 261-273. doi: 10.1016/0377-0427(95)00277-4. [32] J. Zhang and Z. Liu, A further study on inverse linear programming problems, Journal of Computational and Applied Mathematics, 106 (1999), 345-359. doi: 10.1016/S0377-0427(99)00080-1. [33] J. Zhang, Z. Liu and Z. Ma, Some reverse location problems, European Journal of Operations Research, 124 (2000), 77-88. doi: 10.1016/S0377-2217(99)00122-8. [34] J. Zhang and Z. Ma, Solution structure of some inverse combinatorial optimization problems, Journal of Combinatorial Optimization, 3 (1999), 127-139. doi: 10.1023/A:1009829525096. [35] J. Zhang and L. Zhang, An augmented Lagrangian method for a class of inverse quadratic programming problems, Applied Mathematics and Optimization, 61 (2010), 57-83. doi: 10.1007/s00245-009-9075-z.

show all references

##### References:
 [1] M. Afonso, J. Bioucas-Dias and M. Figueiredo, Fast image recovery using variable splitting and constrained optimization, IEEE Transactions on image processing, 19 (2010), 2345-2356. doi: 10.1109/TIP.2010.2047910. [2] R. Ahuja and J. Orlin, Inverse optimization, Operations Research, 49 (2001), 771-783. doi: 10.1287/opre.49.5.771.10607. [3] R. Ahuja and J. Orlin, Combinatorial algorithms for inverse network flow problems, Networks, 40 (2002), 181-187. doi: 10.1002/net.10048. [4] J. Barzilai and J. M. Borwein, Two point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141. [5] D. Bertsekas, On the Goldstein-Levitin-Polyak gradient projection method, IEEE Transactions on Automatic Control, 21 (1976), 174-184. doi: 10.1109/TAC.1976.1101194. [6] E. Birgin, J. Martínez and M. Raydan, Nonmonotone spectral projected gradient methods on convex sets, SIAM Journal on Optimization, 10 (2000), 1196-1211. doi: 10.1137/S1052623497330963. [7] E. Birgin, J. Martínez and M. Raydan, Spectral projected gradient methods: reviews and perspective,, Available from: , (). [8] S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Foundations and Trends in Machine Learning, 3 (2011), 1-122. doi: 10.1561/2200000016. [9] W. Burton and P. Toint, On an instance of the inverse shortest paths problem, Mathematical Programming, 53 (1992), 45-61. doi: 10.1007/BF01585693. [10] M. Cai, X. Yang and J. Zhang, The complexity analysis of the inverse center location problem, Journal of Global Optimization, 15 (1999), 213-218. doi: 10.1023/A:1008360312607. [11] Y. Dai and L. Liao, R-linear convergence of the Barzilai and Borwein gradient method, IMA Journal on Numerical Analysis, 22 (2002), 1-10. doi: 10.1093/imanum/22.1.1. [12] J. Douglas and H. Rachford, On the numerical solution of the heat conduction problem in two and three space variables, Transactions of the American Mathematical Society, 82 (1956), 421-439. doi: 10.1090/S0002-9947-1956-0084194-4. [13] A. Friedlander, J. M. Martínez, B. Molina and M. Raydan, Gradient method with retards and generalizations, SIAM Journal on Numerical Analysis, 36 (1999), 275-289. doi: 10.1137/S003614299427315X. [14] D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite-element approximations, Computers & Mathematics with Applications, 2 (1976), 17-40. doi: 10.1016/0898-1221(76)90003-1. [15] R. Glowinski and P. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989. doi: 10.1137/1.9781611970838. [16] T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891. [17] C. Heuberger, Inverse combinatorial optimization: A survey on problems, methods and results, Journal of Combinatorial Optimization, 8 (2004), 329-361. doi: 10.1023/B:JOCO.0000038914.26975.9b. [18] B. He, L. Liao, D. Han and H. Yang, A new inexact alternating direction method for monotone variational inequalities, Mathematical Programming, 92 (2002), 103-118. doi: 10.1007/s101070100280. [19] B. He, M. Tao and X. Yuan, Alternating direction method with Gaussian back substitution for separable convex programming, SIAM Journal on Optimization, 22 (2012), 313-340. doi: 10.1137/110822347. [20] G. Iyengar and W. Kang, Inverse conic programming and applications, Operations Research Letters, 33 (2005), 319-330. doi: 10.1016/j.orl.2004.04.007. [21] D. Peaceman and H. Rachford, The numerical solution of parabolic elliptic differential equations, Journal of the Society for Industrial and Applied Mathematics, 3 (1955), 28-41. doi: 10.1137/0103003. [22] M. Raydan, On the Barzilai and Borwein choice of steplength for the gradient method, IMA Journal of Numerical Analysis, 13 (1993), 321-326. doi: 10.1093/imanum/13.3.321. [23] M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM Journal on Optimization, 7 (1997), 26-33. doi: 10.1137/S1052623494266365. [24] R. Rockafellar and R. Wets, Variational Analysis, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-642-02431-3. [25] D. Sun, J. Sun and L. Zhang, The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming, Mathematical Programming, 114 (2008), 349-391. doi: 10.1007/s10107-007-0105-9. [26] P. Tseng, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM Journal on Control and Optimization, 29 (1991), 119-138. doi: 10.1137/0329006. [27] P. Tseng, Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming, Mathematical Programming, 48 (1990), 249-263. doi: 10.1007/BF01582258. [28] X. Xiao, L. Zhang and J. Zhang, A smoothing Newton method for a type of inverse semi-definite quadratic programming problem, Journal of Computational and Applied Mathematics, 223 (2009), 485-498. doi: 10.1016/j.cam.2008.01.028. [29] X. Xiao, L. Zhang and J. Zhang, On convergence of augmented Lagrange method for inverse semi-definite quadratic programming problems, Journal of Industrial and Management Optimization, 5 (2009), 319-339. doi: 10.3934/jimo.2009.5.319. [30] J. Yang and Y. Zhang, Alternating direction algorithms for $l_1$-problems in compressive sensing, SIAM Journal on Scientific Computing, 33 (2011), 250-278. doi: 10.1137/090777761. [31] J. Zhang and Z. Liu, Calculating some inverse linear programming problems, Journal of Computational and Applied Mathematics, 72 (1996), 261-273. doi: 10.1016/0377-0427(95)00277-4. [32] J. Zhang and Z. Liu, A further study on inverse linear programming problems, Journal of Computational and Applied Mathematics, 106 (1999), 345-359. doi: 10.1016/S0377-0427(99)00080-1. [33] J. Zhang, Z. Liu and Z. Ma, Some reverse location problems, European Journal of Operations Research, 124 (2000), 77-88. doi: 10.1016/S0377-2217(99)00122-8. [34] J. Zhang and Z. Ma, Solution structure of some inverse combinatorial optimization problems, Journal of Combinatorial Optimization, 3 (1999), 127-139. doi: 10.1023/A:1009829525096. [35] J. Zhang and L. Zhang, An augmented Lagrangian method for a class of inverse quadratic programming problems, Applied Mathematics and Optimization, 61 (2010), 57-83. doi: 10.1007/s00245-009-9075-z.
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