American Institute of Mathematical Sciences

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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), 401-426. 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), 235-263. Google Scholar

[3]

Y. Gao, "Structured Low Rank Matrix Optimization Problems: A Penalty Approach," Ph.D thesis, National University of Singapore, 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), 1432-1457. doi: 10.1137/080727075.  Google Scholar

[5]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21, April, 2011. Available from: http://cvxr.com/cvx. 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), 617-630. doi: 10.1287/opre.1100.0910.  Google Scholar

[7]

R. Horst and N. V. Thoni, DC programming: Overview, J. Optim. Theory Appl., 103 (1999), 1-43.  Google Scholar

[8]

D. Klatte and W. Li, Asymptotic constraint qualifications and global error bounds for convex inequalities, Math. Program., 84 (1999), 137-160.  Google Scholar

[9]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. Optim., 17 (2006), 969-996. doi: 10.1137/050622328.  Google Scholar

[10]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[11]

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

[12]

A. Shapiro, D. Dentcheva and A. Ruszczyński, "Lectures on Stochastic Programming: Modeling and Theory," MPS/SIAM Series on Optimization, 9, SIAM, Philadelphia, PA, 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), 401-426. 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), 235-263. Google Scholar

[3]

Y. Gao, "Structured Low Rank Matrix Optimization Problems: A Penalty Approach," Ph.D thesis, National University of Singapore, 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), 1432-1457. doi: 10.1137/080727075.  Google Scholar

[5]

M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21, April, 2011. Available from: http://cvxr.com/cvx. 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), 617-630. doi: 10.1287/opre.1100.0910.  Google Scholar

[7]

R. Horst and N. V. Thoni, DC programming: Overview, J. Optim. Theory Appl., 103 (1999), 1-43.  Google Scholar

[8]

D. Klatte and W. Li, Asymptotic constraint qualifications and global error bounds for convex inequalities, Math. Program., 84 (1999), 137-160.  Google Scholar

[9]

A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. Optim., 17 (2006), 969-996. doi: 10.1137/050622328.  Google Scholar

[10]

R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[11]

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

[12]

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

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