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

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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

Abstract
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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 and 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.
[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.
[3]	Y. Gao, "Structured Low Rank Matrix Optimization Problems: A Penalty Approach," Ph.D thesis, National University of Singapore, 2010.
[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.
[5]	M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21, April, 2011. Available from: http://cvxr.com/cvx.
[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.
[7]	R. Horst and N. V. Thoni, DC programming: Overview, J. Optim. Theory Appl., 103 (1999), 1-43.
[8]	D. Klatte and W. Li, Asymptotic constraint qualifications and global error bounds for convex inequalities, Math. Program., 84 (1999), 137-160.
[9]	A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. Optim., 17 (2006), 969-996. doi: 10.1137/050622328.
[10]	R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.
[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.
[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.

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]	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.
[3]	Y. Gao, "Structured Low Rank Matrix Optimization Problems: A Penalty Approach," Ph.D thesis, National University of Singapore, 2010.
[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.
[5]	M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21, April, 2011. Available from: http://cvxr.com/cvx.
[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.
[7]	R. Horst and N. V. Thoni, DC programming: Overview, J. Optim. Theory Appl., 103 (1999), 1-43.
[8]	D. Klatte and W. Li, Asymptotic constraint qualifications and global error bounds for convex inequalities, Math. Program., 84 (1999), 137-160.
[9]	A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. Optim., 17 (2006), 969-996. doi: 10.1137/050622328.
[10]	R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970.
[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.
[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.

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