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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 & 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.  doi: 10.1109/TIP.2010.2047910.  Google Scholar [2] R. Ahuja and J. Orlin, Inverse optimization,, Operations Research, 49 (2001), 771.  doi: 10.1287/opre.49.5.771.10607.  Google Scholar [3] R. Ahuja and J. Orlin, Combinatorial algorithms for inverse network flow problems,, Networks, 40 (2002), 181.  doi: 10.1002/net.10048.  Google Scholar [4] J. Barzilai and J. M. Borwein, Two point step size gradient methods,, IMA Journal of Numerical Analysis, 8 (1988), 141.  doi: 10.1093/imanum/8.1.141.  Google Scholar [5] D. Bertsekas, On the Goldstein-Levitin-Polyak gradient projection method,, IEEE Transactions on Automatic Control, 21 (1976), 174.  doi: 10.1109/TAC.1976.1101194.  Google Scholar [6] E. Birgin, J. Martínez and M. Raydan, Nonmonotone spectral projected gradient methods on convex sets,, SIAM Journal on Optimization, 10 (2000), 1196.  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] 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.  doi: 10.1561/2200000016.  Google Scholar [9] W. Burton and P. Toint, On an instance of the inverse shortest paths problem,, Mathematical Programming, 53 (1992), 45.  doi: 10.1007/BF01585693.  Google Scholar [10] M. Cai, X. Yang and J. Zhang, The complexity analysis of the inverse center location problem,, Journal of Global Optimization, 15 (1999), 213.  doi: 10.1023/A:1008360312607.  Google Scholar [11] Y. Dai and L. Liao, R-linear convergence of the Barzilai and Borwein gradient method,, IMA Journal on Numerical Analysis, 22 (2002), 1.  doi: 10.1093/imanum/22.1.1.  Google Scholar [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.  doi: 10.1090/S0002-9947-1956-0084194-4.  Google Scholar [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.  doi: 10.1137/S003614299427315X.  Google Scholar [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.  doi: 10.1016/0898-1221(76)90003-1.  Google Scholar [15] R. Glowinski and P. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM, (1989).  doi: 10.1137/1.9781611970838.  Google Scholar [16] T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323.  doi: 10.1137/080725891.  Google Scholar [17] C. Heuberger, Inverse combinatorial optimization: A survey on problems, methods and results,, Journal of Combinatorial Optimization, 8 (2004), 329.  doi: 10.1023/B:JOCO.0000038914.26975.9b.  Google Scholar [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.  doi: 10.1007/s101070100280.  Google Scholar [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.  doi: 10.1137/110822347.  Google Scholar [20] G. Iyengar and W. Kang, Inverse conic programming and applications,, Operations Research Letters, 33 (2005), 319.  doi: 10.1016/j.orl.2004.04.007.  Google Scholar [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.  doi: 10.1137/0103003.  Google Scholar [22] M. Raydan, On the Barzilai and Borwein choice of steplength for the gradient method,, IMA Journal of Numerical Analysis, 13 (1993), 321.  doi: 10.1093/imanum/13.3.321.  Google Scholar [23] M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem,, SIAM Journal on Optimization, 7 (1997), 26.  doi: 10.1137/S1052623494266365.  Google Scholar [24] R. Rockafellar and R. Wets, Variational Analysis,, Springer-Verlag, (1998).  doi: 10.1007/978-3-642-02431-3.  Google Scholar [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.  doi: 10.1007/s10107-007-0105-9.  Google Scholar [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.  doi: 10.1137/0329006.  Google Scholar [27] P. Tseng, Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming,, Mathematical Programming, 48 (1990), 249.  doi: 10.1007/BF01582258.  Google Scholar [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.  doi: 10.1016/j.cam.2008.01.028.  Google Scholar [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.  doi: 10.3934/jimo.2009.5.319.  Google Scholar [30] J. Yang and Y. Zhang, Alternating direction algorithms for $l_1$-problems in compressive sensing,, SIAM Journal on Scientific Computing, 33 (2011), 250.  doi: 10.1137/090777761.  Google Scholar [31] J. Zhang and Z. Liu, Calculating some inverse linear programming problems,, Journal of Computational and Applied Mathematics, 72 (1996), 261.  doi: 10.1016/0377-0427(95)00277-4.  Google Scholar [32] J. Zhang and Z. Liu, A further study on inverse linear programming problems,, Journal of Computational and Applied Mathematics, 106 (1999), 345.  doi: 10.1016/S0377-0427(99)00080-1.  Google Scholar [33] J. Zhang, Z. Liu and Z. Ma, Some reverse location problems,, European Journal of Operations Research, 124 (2000), 77.  doi: 10.1016/S0377-2217(99)00122-8.  Google Scholar [34] J. Zhang and Z. Ma, Solution structure of some inverse combinatorial optimization problems,, Journal of Combinatorial Optimization, 3 (1999), 127.  doi: 10.1023/A:1009829525096.  Google Scholar [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.  doi: 10.1007/s00245-009-9075-z.  Google Scholar

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.  doi: 10.1109/TIP.2010.2047910.  Google Scholar [2] R. Ahuja and J. Orlin, Inverse optimization,, Operations Research, 49 (2001), 771.  doi: 10.1287/opre.49.5.771.10607.  Google Scholar [3] R. Ahuja and J. Orlin, Combinatorial algorithms for inverse network flow problems,, Networks, 40 (2002), 181.  doi: 10.1002/net.10048.  Google Scholar [4] J. Barzilai and J. M. Borwein, Two point step size gradient methods,, IMA Journal of Numerical Analysis, 8 (1988), 141.  doi: 10.1093/imanum/8.1.141.  Google Scholar [5] D. Bertsekas, On the Goldstein-Levitin-Polyak gradient projection method,, IEEE Transactions on Automatic Control, 21 (1976), 174.  doi: 10.1109/TAC.1976.1101194.  Google Scholar [6] E. Birgin, J. Martínez and M. Raydan, Nonmonotone spectral projected gradient methods on convex sets,, SIAM Journal on Optimization, 10 (2000), 1196.  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] 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.  doi: 10.1561/2200000016.  Google Scholar [9] W. Burton and P. Toint, On an instance of the inverse shortest paths problem,, Mathematical Programming, 53 (1992), 45.  doi: 10.1007/BF01585693.  Google Scholar [10] M. Cai, X. Yang and J. Zhang, The complexity analysis of the inverse center location problem,, Journal of Global Optimization, 15 (1999), 213.  doi: 10.1023/A:1008360312607.  Google Scholar [11] Y. Dai and L. Liao, R-linear convergence of the Barzilai and Borwein gradient method,, IMA Journal on Numerical Analysis, 22 (2002), 1.  doi: 10.1093/imanum/22.1.1.  Google Scholar [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.  doi: 10.1090/S0002-9947-1956-0084194-4.  Google Scholar [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.  doi: 10.1137/S003614299427315X.  Google Scholar [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.  doi: 10.1016/0898-1221(76)90003-1.  Google Scholar [15] R. Glowinski and P. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM, (1989).  doi: 10.1137/1.9781611970838.  Google Scholar [16] T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems,, SIAM Journal on Imaging Sciences, 2 (2009), 323.  doi: 10.1137/080725891.  Google Scholar [17] C. Heuberger, Inverse combinatorial optimization: A survey on problems, methods and results,, Journal of Combinatorial Optimization, 8 (2004), 329.  doi: 10.1023/B:JOCO.0000038914.26975.9b.  Google Scholar [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.  doi: 10.1007/s101070100280.  Google Scholar [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.  doi: 10.1137/110822347.  Google Scholar [20] G. Iyengar and W. Kang, Inverse conic programming and applications,, Operations Research Letters, 33 (2005), 319.  doi: 10.1016/j.orl.2004.04.007.  Google Scholar [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.  doi: 10.1137/0103003.  Google Scholar [22] M. Raydan, On the Barzilai and Borwein choice of steplength for the gradient method,, IMA Journal of Numerical Analysis, 13 (1993), 321.  doi: 10.1093/imanum/13.3.321.  Google Scholar [23] M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem,, SIAM Journal on Optimization, 7 (1997), 26.  doi: 10.1137/S1052623494266365.  Google Scholar [24] R. Rockafellar and R. Wets, Variational Analysis,, Springer-Verlag, (1998).  doi: 10.1007/978-3-642-02431-3.  Google Scholar [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.  doi: 10.1007/s10107-007-0105-9.  Google Scholar [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.  doi: 10.1137/0329006.  Google Scholar [27] P. Tseng, Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming,, Mathematical Programming, 48 (1990), 249.  doi: 10.1007/BF01582258.  Google Scholar [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.  doi: 10.1016/j.cam.2008.01.028.  Google Scholar [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.  doi: 10.3934/jimo.2009.5.319.  Google Scholar [30] J. Yang and Y. Zhang, Alternating direction algorithms for $l_1$-problems in compressive sensing,, SIAM Journal on Scientific Computing, 33 (2011), 250.  doi: 10.1137/090777761.  Google Scholar [31] J. Zhang and Z. Liu, Calculating some inverse linear programming problems,, Journal of Computational and Applied Mathematics, 72 (1996), 261.  doi: 10.1016/0377-0427(95)00277-4.  Google Scholar [32] J. Zhang and Z. Liu, A further study on inverse linear programming problems,, Journal of Computational and Applied Mathematics, 106 (1999), 345.  doi: 10.1016/S0377-0427(99)00080-1.  Google Scholar [33] J. Zhang, Z. Liu and Z. Ma, Some reverse location problems,, European Journal of Operations Research, 124 (2000), 77.  doi: 10.1016/S0377-2217(99)00122-8.  Google Scholar [34] J. Zhang and Z. Ma, Solution structure of some inverse combinatorial optimization problems,, Journal of Combinatorial Optimization, 3 (1999), 127.  doi: 10.1023/A:1009829525096.  Google Scholar [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.  doi: 10.1007/s00245-009-9075-z.  Google Scholar
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