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October  2016, 12(4): 1349-1366. doi: 10.3934/jimo.2016.12.1349

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

Received  September 2014 Revised  May 2015 Published  January 2016

We consider an optimization problem that minimizes a function of the form $f=f_0+f_1-f_2$ with the constraint $g-h\leq 0$, where $f_0$ is continuous differentiable, $f_1,f_2$ are convex and $g,h$ are lower semicontinuous convex. We propose to solve the problem by an inexact subgradient-based convex approximations method. Under mild assumptions, we show that the method is guaranteed to converge to a stationary point. Finally, some preliminary numerical results are given.
Citation: Jian Gu, Xiantao Xiao, Liwei Zhang. A subgradient-based convex approximations method for DC programming and its applications. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1349-1366. doi: 10.3934/jimo.2016.12.1349
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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