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A subgradient-based convex approximations method for DC programming and its applications
1. | School of Sciences, Dalian Ocean University, Dalian 116023, China |
2. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116023 |
References:
[1] |
L. T. H. An, An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints, Math. Program., 87 (2000), 401-426.
doi: 10.1007/s101070050003. |
[2] |
L. T. H. An, L. H. Minh and P. D. Tao, Optimization based DC programming and DCA for hierarchical clustering, European J. Oper. Res., 183 (2007), 1067-1085.
doi: 10.1016/j.ejor.2005.07.028. |
[3] |
L. T. H. An and P. D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Ann. Oper. Res., 133 (2005), 23-46.
doi: 10.1007/s10479-004-5022-1. |
[4] |
C. Audet, P. Hansen, B. Jaumard and G. Savard, A branch and cut algorithm for nonconvex quadratically constrained quadratic programming, Math. Program., 87 (2000), 131-152. |
[5] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, {Springer, New York}, 2000.
doi: 10.1007/978-1-4612-1394-9. |
[6] |
D. H. Fang, C. Li and X. Q. Yang, Asymptotic closure condition and Fenchel duality for DC optimization problems in locally convex spaces, Nonliner Anal., 75 (2012), 3672-3681.
doi: 10.1016/j.na.2012.01.023. |
[7] |
M. Fazel, Matrix Rank Minimization with Applications, PhD thesis, Stanford University, 2002. |
[8] |
Y. Gao, Structured Low Rank Matrix Optimization Problems: A Penalty Approach, PhD thesis, National University of Singapore, 2010. |
[9] |
S. He, Z. Luo, J. Nie and S. Zhang, Semidefinite relaxation bounds for indefinite homogeneous quadratic optimization, SIAM J. Optim., 19 (2008), 503-523.
doi: 10.1137/070679041. |
[10] |
L. J. Hong, Y. Yang and L. Zhang, Sequential convex approximations to joint chance constrained programs: a Monte Carlo approach, Oper. Res., 59 (2011), 617-630.
doi: 10.1287/opre.1100.0910. |
[11] |
R. Horst and N. V. Thoni, DC programming: Overview, J. Optim. Theory Appl., 103 (1999), 1-43.
doi: 10.1023/A:1021765131316. |
[12] |
Z. Luo, W. Ma, A. So, Y. Ye and S. Zhang, Semidefinite relaxation of quadratic optimization problems, IEEE Signal Process Mag., 27 (2010), 20-34.
doi: 10.1109/MSP.2010.936019. |
[13] |
Z. Luo, N. Sidiropoulos, P. Tseng and S. Zhang, Approximation bounds for quadratic optimization with homogeneous quadratic constraints, SIAM J. Optim., 18 (2007), 1-28.
doi: 10.1137/050642691. |
[14] |
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I: Basic Theory, Springer, Berlin, 2006. |
[15] |
A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. Optim., 17 (2006), 969-996.
doi: 10.1137/050622328. |
[16] |
B. Recht, M. Fazel and P. A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), 471-501.
doi: 10.1137/070697835. |
[17] |
R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer, New York, 1998.
doi: 10.1007/978-3-642-02431-3. |
[18] |
W. Schirotzek, Nonsmooth Analysis, Springer, Berlin, 2007.
doi: 10.1007/978-3-540-71333-3. |
[19] |
F. Shan, L. Zhang and X. Xiao, A smoothing function approach to joint chance-constrained programs, J. Optim. Theory Appl., 59 (2014), 181-199.
doi: 10.1007/s10957-013-0513-3. |
[20] |
A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2009.
doi: 10.1137/1.9780898718751. |
[21] |
N. Sidiropoulos, T. Davidson and Z. Luo, Transmit beamforming for physical-layer multicasting, IEEE Trans. Signal Process., 54 (2006), 2239-2251.
doi: 10.1109/TSP.2006.872578. |
[22] |
N. V. Thoai, Reverse convex programming approach in the space of extreme criteria for optimization over efficient sets, J. Optim. Theory Appl., 147 (2010), 263-277.
doi: 10.1007/s10957-010-9721-2. |
[23] |
X. Xiao, J. Gu, L. Zhang and S. Zhang, A sequential convex program method to DC program with joint chance constraints, J. Ind. Manag. Optim., 8 (2012), 733-747.
doi: 10.3934/jimo.2012.8.733. |
show all references
References:
[1] |
L. T. H. An, An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints, Math. Program., 87 (2000), 401-426.
doi: 10.1007/s101070050003. |
[2] |
L. T. H. An, L. H. Minh and P. D. Tao, Optimization based DC programming and DCA for hierarchical clustering, European J. Oper. Res., 183 (2007), 1067-1085.
doi: 10.1016/j.ejor.2005.07.028. |
[3] |
L. T. H. An and P. D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems, Ann. Oper. Res., 133 (2005), 23-46.
doi: 10.1007/s10479-004-5022-1. |
[4] |
C. Audet, P. Hansen, B. Jaumard and G. Savard, A branch and cut algorithm for nonconvex quadratically constrained quadratic programming, Math. Program., 87 (2000), 131-152. |
[5] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, {Springer, New York}, 2000.
doi: 10.1007/978-1-4612-1394-9. |
[6] |
D. H. Fang, C. Li and X. Q. Yang, Asymptotic closure condition and Fenchel duality for DC optimization problems in locally convex spaces, Nonliner Anal., 75 (2012), 3672-3681.
doi: 10.1016/j.na.2012.01.023. |
[7] |
M. Fazel, Matrix Rank Minimization with Applications, PhD thesis, Stanford University, 2002. |
[8] |
Y. Gao, Structured Low Rank Matrix Optimization Problems: A Penalty Approach, PhD thesis, National University of Singapore, 2010. |
[9] |
S. He, Z. Luo, J. Nie and S. Zhang, Semidefinite relaxation bounds for indefinite homogeneous quadratic optimization, SIAM J. Optim., 19 (2008), 503-523.
doi: 10.1137/070679041. |
[10] |
L. J. Hong, Y. Yang and L. Zhang, Sequential convex approximations to joint chance constrained programs: a Monte Carlo approach, Oper. Res., 59 (2011), 617-630.
doi: 10.1287/opre.1100.0910. |
[11] |
R. Horst and N. V. Thoni, DC programming: Overview, J. Optim. Theory Appl., 103 (1999), 1-43.
doi: 10.1023/A:1021765131316. |
[12] |
Z. Luo, W. Ma, A. So, Y. Ye and S. Zhang, Semidefinite relaxation of quadratic optimization problems, IEEE Signal Process Mag., 27 (2010), 20-34.
doi: 10.1109/MSP.2010.936019. |
[13] |
Z. Luo, N. Sidiropoulos, P. Tseng and S. Zhang, Approximation bounds for quadratic optimization with homogeneous quadratic constraints, SIAM J. Optim., 18 (2007), 1-28.
doi: 10.1137/050642691. |
[14] |
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I: Basic Theory, Springer, Berlin, 2006. |
[15] |
A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. Optim., 17 (2006), 969-996.
doi: 10.1137/050622328. |
[16] |
B. Recht, M. Fazel and P. A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), 471-501.
doi: 10.1137/070697835. |
[17] |
R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer, New York, 1998.
doi: 10.1007/978-3-642-02431-3. |
[18] |
W. Schirotzek, Nonsmooth Analysis, Springer, Berlin, 2007.
doi: 10.1007/978-3-540-71333-3. |
[19] |
F. Shan, L. Zhang and X. Xiao, A smoothing function approach to joint chance-constrained programs, J. Optim. Theory Appl., 59 (2014), 181-199.
doi: 10.1007/s10957-013-0513-3. |
[20] |
A. Shapiro, D. Dentcheva and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, SIAM, Philadelphia, 2009.
doi: 10.1137/1.9780898718751. |
[21] |
N. Sidiropoulos, T. Davidson and Z. Luo, Transmit beamforming for physical-layer multicasting, IEEE Trans. Signal Process., 54 (2006), 2239-2251.
doi: 10.1109/TSP.2006.872578. |
[22] |
N. V. Thoai, Reverse convex programming approach in the space of extreme criteria for optimization over efficient sets, J. Optim. Theory Appl., 147 (2010), 263-277.
doi: 10.1007/s10957-010-9721-2. |
[23] |
X. Xiao, J. Gu, L. Zhang and S. Zhang, A sequential convex program method to DC program with joint chance constraints, J. Ind. Manag. Optim., 8 (2012), 733-747.
doi: 10.3934/jimo.2012.8.733. |
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