Abstract
Many important practical problems can be formulated as probabilistic constrained optimization problem (PCOP), which is challenging to solve since it is usually non-convex and non-smooth. Effective methods for (PCOP) mostly focus on approximation techniques. This paper aims at studying the D.C. (difference of two convex functions) approximation techniques. A D.C. approximation is explored to solve the probabilistic constrained optimization problem based on Chen–Harker–Kanzow–Smale (CHKS) smooth plus function. A smooth approximation to probabilistic constraint function is proposed and the corresponding D.C. approximation problem is established. It is proved that the approximation problem is equivalent to the original one under certain conditions. Sequential convex approximation (SCA) algorithm is implemented to solve the D.C. approximation problem. Sample average approximation method is applied to solve the convex subproblem. Numerical results suggest that D.C. approximation technique is effective for optimization with probabilistic constraints.
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References
Ahmed S, Luedtke J, Song Y, Xie W (2017) Nonanticipative duality, relaxations, and formulations for chance-constrained stochastic programs. Math Program 162:51–81
Alejandra P, Luedtke J, Wachter A (2020) Solving chance constrained problems via a smooth sample-based nonlinear approximation. SIAM J Optim 30(3):2221–2250
Ben-Tal A, Nemirovski A (2000) Robust solutions of linear programming problems contaminated with uncertain data. Math Program 88:411–424
Calafiore G, Campi MC (2005) Uncertain convex programs: randomized solutions and confidence levels. Math Program 102:25–46
Calafiore G, Campi MC (2006) The scenario approach to robust control design. IEEE Trans Automat Contr 51:742–753
Charnes A, Cooper WW, Symonds H (1958) Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag Sci 4:235–263
Chen C (1996) A class of smoothing functions for nonlinear and mixed complementarity problems. Comput Optim Appl 5:97–138
Farias DPD, Roy BV (2004) On constraint sampling in the linear programming approach to approximate dynamic programming. Math Oper Res 29:462–478
Hong LJ, Yang Y, Zhang LW (2011) Sequential convex approximations to joint chance constrained programs: a Monte Carlo approach. Oper Res 59:617–630
Hu Z, Hong LJ, Zhang LW (2013) A smooth Monte Carlo approach to joint chance-constrained programs. IIE Trans 45:716–735
Jiang N, Xie W (2022) ALSO-X and ALSO-X+: better convex approximations for chance constrained programs. Oper Res 70(6):3581–3600
Luedtke J, Ahmed S (2008) A sample approximation approach for optimization with probabilistic constraints. SIAM J Optim 19:674–699
Miller LB, Wagner HM (1965) Chance constrained programming with joint constraints. Oper Res 13:930–945
Nemirovski A, Shapiro A (2006) Convex approximations of chance constrained programs. SIAM J Optim 17:347–375
Pagnonclli BK, Ahmed S, Shapiro A (2009) Sample average approximation method for chance constrained programming: theory and applications. J OPtim Theory Appl 142:399–416
Ren Y, Xiong Y, Yan Y, Gu J (2022) A smooth approximation approach for optimization with probabilistic constraints based on Sigmoid function. J Inequal Appl 38:1–14
Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21–41
Rockafellar RT, Wets RJB (1998) Variational analysis. Springer, Berlin
Rohit K, Luedtke J (2021) A stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Math Program Comput 13(4):705–751
Shan F, Zhang LW, Xiao XT (2014) A smoothing function approach to joint chance-constrained programs. J Optim Theory Appl 163:181–199
Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming: modeling and theory. Society for Industrial and Applied Mathematics, Philadelphia
Yang Y, Sutanto C (2019) Chance-constrained optimization for nonconvex programs using scenario-based methods. ISA Trans 90:157–168
Acknowledgements
We would like to thank the constructive feedback provided by the reviewers.
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Our work is supported by National Natural Science Foundation of China under Grant (12171219); Liaoning Province Department of Education Scientific Research General Project (LJ2019005).
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YR performed the analysis with constructive discussions and wrote the manuscript. YS and DL performed the experiments. FG contributed significantly to data analysis. All authors have read and approved the manuscript.
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Ren, Y., Sun, Y., Li, D. et al. A D.C. approximation approach for optimization with probabilistic constraints based on Chen–Harker–Kanzow–Smale smooth plus function. Math Meth Oper Res (2024). https://doi.org/10.1007/s00186-024-00859-y
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DOI: https://doi.org/10.1007/s00186-024-00859-y