American Institute of Mathematical Sciences

Advanced
July  2012, 8(3): 733-747. doi: 10.3934/jimo.2012.8.733

A sequential convex program method to DC program with joint chance constraints

Xiantao Xiao 1, , Jian Gu 2, , Liwei Zhang 1, and  Shaowu Zhang 3,

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116023, China, China
2. 

School of Sciences, Dalian Ocean University, Dalian 116023, China
3. 

School of Computer Science and Technology, Dalian University of Technology, Dalian 116023, China

Received  September 2011 Revised  March 2012 Published  June 2012

In this paper, we consider a DC (difference of convex) programming problem with joint chance constraints (JCCDCP). We propose a DC function to approximate the constrained function and a corresponding DC program ($\textrm{P}_{\varepsilon}$) to approximate the JCCDCP. Under some mild assumptions, we show that the solution of Problem ($\textrm{P}_{\varepsilon}$) converges to the solution of JCCDCP when $\varepsilon\downarrow 0$. A sequential convex program method is constructed to solve the Problem ($\textrm{P}_{\varepsilon}$). At each iteration a convex program is solved based on the Monte Carlo method, and the generated optimal sequence is proved to converge to the stationary point of Problem ($\textrm{P}_{\varepsilon}$).
Keywords: joint chance constraints., DC programming, Sequential convex approximation approach.
Mathematics Subject Classification: Primary: 65K05, 90C26, 90C1.
Citation: Xiantao Xiao, Jian Gu, Liwei Zhang, Shaowu Zhang. A sequential convex program method to DC program with joint chance constraints. Journal of Industrial & Management Optimization, 2012, 8 (3) : 733-747. doi: 10.3934/jimo.2012.8.733
References:
[1]

L. T. H. An, An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints,, Math. Program., 87 (2000), 401. doi: 10.1007/s101070050003. Google Scholar

[2]

A. Charnes, W. W. Cooper and H. Symonds, Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil,, Manag. Sci., 4 (1958), 235. Google Scholar

[3]

Y. Gao, "Structured Low Rank Matrix Optimization Problems: A Penalty Approach,", Ph.D thesis, (2010). Google Scholar

[4]

Y. Gao and D. Sun, Calibrating least squares semidefinite programming with equality and inequality constraints,, SIAM J. Matrix Anal. Appl., 31 (2009), 1432. doi: 10.1137/080727075. Google Scholar

[5]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21,, April, (2011). Google Scholar

[6]

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. doi: 10.1287/opre.1100.0910. Google Scholar

[7]

R. Horst and N. V. Thoni, DC programming: Overview,, J. Optim. Theory Appl., 103 (1999), 1. Google Scholar

[8]

D. Klatte and W. Li, Asymptotic constraint qualifications and global error bounds for convex inequalities,, Math. Program., 84 (1999), 137. Google Scholar

[9]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs,, SIAM J. Optim., 17 (2006), 969. doi: 10.1137/050622328. Google Scholar

[10]

R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970). Google Scholar

[11]

R. T. Rockafellar and R. J. B. Wets, "Variational Analysis,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317 (1998). Google Scholar

[12]

A. Shapiro, D. Dentcheva and A. Ruszczyński, "Lectures on Stochastic Programming: Modeling and Theory,", MPS/SIAM Series on Optimization, 9 (2009). Google Scholar

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. doi: 10.1007/s101070050003. Google Scholar

[2]

A. Charnes, W. W. Cooper and H. Symonds, Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil,, Manag. Sci., 4 (1958), 235. Google Scholar

[3]

Y. Gao, "Structured Low Rank Matrix Optimization Problems: A Penalty Approach,", Ph.D thesis, (2010). Google Scholar

[4]

Y. Gao and D. Sun, Calibrating least squares semidefinite programming with equality and inequality constraints,, SIAM J. Matrix Anal. Appl., 31 (2009), 1432. doi: 10.1137/080727075. Google Scholar

[5]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21,, April, (2011). Google Scholar

[6]

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. doi: 10.1287/opre.1100.0910. Google Scholar

[7]

R. Horst and N. V. Thoni, DC programming: Overview,, J. Optim. Theory Appl., 103 (1999), 1. Google Scholar

[8]

D. Klatte and W. Li, Asymptotic constraint qualifications and global error bounds for convex inequalities,, Math. Program., 84 (1999), 137. Google Scholar

[9]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs,, SIAM J. Optim., 17 (2006), 969. doi: 10.1137/050622328. Google Scholar

[10]

R. T. Rockafellar, "Convex Analysis,", Princeton Mathematical Series, (1970). Google Scholar

[11]

R. T. Rockafellar and R. J. B. Wets, "Variational Analysis,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317 (1998). Google Scholar

[12]

A. Shapiro, D. Dentcheva and A. Ruszczyński, "Lectures on Stochastic Programming: Modeling and Theory,", MPS/SIAM Series on Optimization, 9 (2009). Google Scholar

[1]

Radu Ioan Boţ, Anca Grad, Gert Wanka. Sequential characterization of solutions in convex composite programming and applications to vector optimization. Journal of Industrial & Management Optimization, 2008, 4 (4) : 767-782. doi: 10.3934/jimo.2008.4.767
[2]

Jian Gu, Xiantao Xiao, Liwei Zhang. A subgradient-based convex approximations method for DC programming and its applications. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1349-1366. doi: 10.3934/jimo.2016.12.1349
[3]

Changzhi Wu, Chaojie Li, Qiang Long. A DC programming approach for sensor network localization with uncertainties in anchor positions. Journal of Industrial & Management Optimization, 2014, 10 (3) : 817-826. doi: 10.3934/jimo.2014.10.817
[4]

Jean-François Babadjian, Clément Mifsud, Nicolas Seguin. Relaxation approximation of Friedrichs' systems under convex constraints. Networks & Heterogeneous Media, 2016, 11 (2) : 223-237. doi: 10.3934/nhm.2016.11.223
[5]

Le Thi Hoai An, Tran Duc Quynh, Pham Dinh Tao. A DC programming approach for a class of bilevel programming problems and its application in Portfolio Selection. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 167-185. doi: 10.3934/naco.2012.2.167
[6]

Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial & Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004
[7]

Zhi-Bin Deng, Ye Tian, Cheng Lu, Wen-Xun Xing. Globally solving quadratic programs with convex objective and complementarity constraints via completely positive programming. Journal of Industrial & Management Optimization, 2018, 14 (2) : 625-636. doi: 10.3934/jimo.2017064
[8]

Ailing Zhang, Shunsuke Hayashi. Celis-Dennis-Tapia based approach to quadratic fractional programming problems with two quadratic constraints. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 83-98. doi: 10.3934/naco.2011.1.83
[9]

Songqiang Qiu, Zhongwen Chen. An adaptively regularized sequential quadratic programming method for equality constrained optimization. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019075
[10]

Yuying Zhou, Gang Li. The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 9-23. doi: 10.3934/naco.2014.4.9
[11]

Yanjun Wang, Kaiji Shen. A new concave reformulation and its application in solving DC programming globally under uncertain environment. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019057
[12]

Gang Li, Xiaoqi Yang, Yuying Zhou. Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces. Journal of Industrial & Management Optimization, 2013, 9 (3) : 671-687. doi: 10.3934/jimo.2013.9.671
[13]

Harald Held, Gabriela Martinez, Philipp Emanuel Stelzig. Stochastic programming approach for energy management in electric microgrids. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 241-267. doi: 10.3934/naco.2014.4.241
[14]

Yingjie Li, Xiaoguang Yang, Shushang Zhu, Dong-Hui Li. A hybrid approach for index tracking with practical constraints. Journal of Industrial & Management Optimization, 2014, 10 (3) : 905-927. doi: 10.3934/jimo.2014.10.905
[15]

Reetabrata Mookherjee, Benjamin F. Hobbs, Terry L. Friesz, Matthew A. Rigdon. Dynamic oligopolistic competition on an electric power network with ramping costs and joint sales constraints. Journal of Industrial & Management Optimization, 2008, 4 (3) : 425-452. doi: 10.3934/jimo.2008.4.425
[16]

Franco Maceri, Michele Marino, Giuseppe Vairo. Equilibrium and stability of tensegrity structures: A convex analysis approach. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 461-478. doi: 10.3934/dcdss.2013.6.461
[17]

Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019033
[18]

Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357
[19]

Ingrid Daubechies, Gerd Teschke, Luminita Vese. Iteratively solving linear inverse problems under general convex constraints. Inverse Problems & Imaging, 2007, 1 (1) : 29-46. doi: 10.3934/ipi.2007.1.29
[20]

Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial & Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences