# American Institute of Mathematical Sciences

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

 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 ($\textrm{P}_{\varepsilon}$).
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.

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##### 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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