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Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback
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 |
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. |
[2] |
R. Ahuja and J. Orlin, Inverse optimization,, Operations Research, 49 (2001), 771.
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.
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.
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.
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.
doi: 10.1137/S1052623497330963. |
[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. |
[9] |
W. Burton and P. Toint, On an instance of the inverse shortest paths problem,, Mathematical Programming, 53 (1992), 45.
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.
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.
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.
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.
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.
doi: 10.1016/0898-1221(76)90003-1. |
[15] |
R. Glowinski and P. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM, (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.
doi: 10.1137/080725891. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
R. Rockafellar and R. Wets, Variational Analysis,, Springer-Verlag, (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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
doi: 10.1109/TIP.2010.2047910. |
[2] |
R. Ahuja and J. Orlin, Inverse optimization,, Operations Research, 49 (2001), 771.
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.
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.
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.
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.
doi: 10.1137/S1052623497330963. |
[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. |
[9] |
W. Burton and P. Toint, On an instance of the inverse shortest paths problem,, Mathematical Programming, 53 (1992), 45.
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.
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.
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.
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.
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.
doi: 10.1016/0898-1221(76)90003-1. |
[15] |
R. Glowinski and P. Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM, (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.
doi: 10.1137/080725891. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
R. Rockafellar and R. Wets, Variational Analysis,, Springer-Verlag, (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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
doi: 10.1007/s00245-009-9075-z. |
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