Abstract
In this article, we consider a DC (difference of two convex functions) function approach for solving joint chance-constrained programs (JCCP), which was first established by Hong et al. (Oper Res 59:617–630, 2011). They used a DC function to approximate the probability function and constructed a sequential convex approximation method to solve the approximation problem. However, the DC function they used was nondifferentiable. To alleviate this difficulty, we propose a class of smoothing functions to approximate the joint chance-constraint function, based on which smooth optimization problems are constructed to approximate JCCP. We show that the solutions of a sequence of smoothing approximations converge to a Karush–Kuhn–Tucker point of JCCP under a certain asymptotic regime. To implement the proposed method, four examples in the class of smoothing functions are explored. Moreover, the numerical experiments show that our method is comparable and effective.
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Notes
Note that the stopping criteria \(\mathrm dist (0,T_{x^k}(x^{k+1}))\le \varepsilon \Vert x^{k+1}-x^{k}\Vert \) is not always easy to verify. It depends on the structure of problem CP(\(x^k\)). Anyway, in practice, any other stopping criteria can be used, so long as \(x^{k+1}\) is an approximation to the KKT point of CP\((x^k)\).
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Acknowledgments
The authors thank the associate editor and two anonymous referees for their helpful comments and suggestions which have greatly improved the quality of this manuscript. This stduy was supported by the National Natural Science Foundation of China under projects No. 11071029, 11101064, and 91130007, and the Fundamental Research Funds for the Central Universities under project No. DUT13LK42.
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Communicated by Jim Luedtke.
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Shan, F., Zhang, L. & Xiao, X. A Smoothing Function Approach to Joint Chance-Constrained Programs. J Optim Theory Appl 163, 181–199 (2014). https://doi.org/10.1007/s10957-013-0513-3
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DOI: https://doi.org/10.1007/s10957-013-0513-3
Keywords
- Joint chance-constrained programs
- Smoothing function
- Sequential convex approximation method
- DC function