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Operations Engineering & Analytics

Chance constrained programs with Gaussian mixture models

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Pages 1117-1130 | Received 06 Apr 2021, Accepted 12 Oct 2021, Published online: 04 Jan 2022

References

  • Bremer, I., Henrion, R. and Möller, A. (2015) Probabilistic constraints via SQP solver: Application to a renewable energy management problem. Computational Management Science, 12(3), 435–459.
  • Bishop, C.M. (2006) Pattern Recognition and Machine Learning, Springer, New York, NY.
  • Buckley, I., Saunders, D. and Seco, L. (2008) Portfolio optimization when asset returns have the Gaussian mixture distribution. European Journal of Operational Research, 185(3), 1434–1461.
  • Calafiore, G. and Campi, M.C. (2005) Uncertain convex programs: Randomized solutions and confidence levels. Mathematical Programming, 102(1), 25–46.
  • Calafiore, G. and Campi, M.C. (2006) The scenario approach to robust control design. IEEE Transactions on Automatic Control, 51(5), 742–753.
  • Charnes, A., Cooper, W.W. and Symonds, G.H. (1958) Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil. Management Science, 4, 235–263.
  • Chen, W., Sim, M., Sun, J. and Teo, C.P. (2010) From CVaR to uncertainty set: Implications in joint chance constrained optimization. Operations Research, 58, 470–485.
  • Chen, Z., Peng, S. and Liu, J. (2018) Data-driven robust chance constrained problems: A mixture model approach. Journal of Optimization Theory and Applications, 179(3), 1065–1085.
  • Dentcheva, D., Prkopa, A. and Ruszczynski, A. (2000) Concavity and efficient points of discrete distributions in probabilistic programming. Mathematical Programming, 89(1), 55–77.
  • Frühwirth-Schnatter, S. (2006) Finite Mixture and Markov Switching Models, Springer, New York, NY.
  • Geng, X. and Xie, L. (2019) Data-driven decision making with probabilistic guarantees (part 1): A schematic overview of chance-constrained optimization. arXiv preprint arXiv:1903.10621.
  • Hanasusanto, G.A., Roitch, V., Kuhn, D. and Wiesemann, W. (2017) Ambiguous joint chance constraints under mean and dispersion information. Operations Research, 65(3), 751–767.
  • Henrion, R. (2007) Structural properties of linear probabilistic constraints. Optimization, 56(4), 425–440.
  • Henrion, R. (2012) Gradient estimates for Gaussian distribution functions: Application to probabilistically constrained optimization problems. Numerical Algebra, Control and Optimization, 2(4), 655–668.
  • Henrion, R. (2013) A critical note on empirical (sample average, Monte Carlo) approximation of solutions to chance constrained programs, in System Modeling and Optimization, volume 391 of IFIP Advances in Information and Communication, Springer-Verlag, Berlin, pp. 25–37.
  • Henrion, R. and Möller, A. (2012) A gradient formula for linear chance constraints under Gaussian distribution. Mathematics of Operations Research, 37(3), 475–488.
  • Hong, L.J., Yang, Y. and Zhang, L. (2011) Sequential convex approximations to joint chance constrained programs: A Monte Carlo approach. Operations Research, 59, 617–630.
  • Horst, R. (1986) A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization. Journal of Optimization Theory and Application, 51, 271–291.
  • Horst, R., Pardalos, P.M. and Thoai, N.V. (1995) Introduction to Global Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands.
  • Hu, Z., Hong, L.J. and Zhang, L. (2013) A smooth Monte Carlo approach to joint chance constrained program. IIE Transactions, 45(7), 716–735.
  • Lasserre, J.B. and Weisser, T. (2021) Distributionally robust polynomial chance-constraints under mixture ambiguity sets. Mathematical Programming, 185, 409–453.
  • Law, A.M. (2013) Simulation Modeling and Analysis, McGraw-Hill, New York, NY.
  • Lejeune, M.A. (2012) Pattern-based modeling and solution of probabilistically constrained optimization problems. Operations Research, 60(6), 1356–1372.
  • Li, J.Q. and Barron, A.R. (2000) Mixture density estimation. Advances in Neural Information Processing Systems, 12, 279–285.
  • Luedtke, J. and Ahmed, S. (2008) A sample approximation approach for optimization with probabilistic constraints. SIAM Journal on Optimization, 19(2), 674–699.
  • Luedtke, J., Ahmed, S. and Nemhauser, G.L. (2010) An integer programming approach for linear programs with probabilistic constraints. Mathematical Programming, 122, 247–272.
  • Maugis-Rabusseau, C. and Michel, B. (2013) Adaptive density estimation for clustering with Gaussian mixtures. ESAIM: Probability and Statistics, 17, 698–724.
  • McLachlan, G.J. and Peel, D. (2000) Finite Mixture Models, Wiley, New York, NY.
  • Miller, L.B. and Wagner, H. (1965) Chance-constrained programming with joint constraints. Operations Research, 13, 930–945.
  • Nemirovski, A. and Shapiro, A. (2006) Convex approximations of chance constrained programs. SIAM Journal on Optimization, 17, 969–996.
  • Pagnoncelli, B.K., Ahmed, S. and Shapiro, A. (2009) Sample average approximation method for chance constrained programming: Theory and applications. Journal of Optimization Theory and Applications, 142, 399–416.
  • Parzen, E. (1962) On estimation of a probability density function and mode. The Annals of Mathematical Statistics, 33(3), 1065–1076.
  • Pearson, K. (1894) Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London, A 185, 71–110.
  • Peña-Ordieres, A., Luedtke, J. and Wächter, A. (2020) Solving chance-constrained problems via a smooth sample-based nonlinear approximation. SIAM Journal on Optimization, 30(3), 2221–2250.
  • Prékopa, A. (2003) Probabilistic programming, in Stochastic Programming, Handbooks in OR&MS. Vol. 10, Elsevier, Amsterdam, The Netherlands, pp. 267–351.
  • Rosenblatt, M. (1956) Remarks on some nonparametric estimates of a density function. The Annals of Mathematical Statistics, 27, 832–837.
  • Sahinidis, N.V. (2017) BARON 17.8.9: Global Optimization of Mixed-Integer Nonlinear Programs, User’s manual. https://sahinidis.coe.gatech.edu/baron
  • Sturm, J.F. (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 11/12, 625–653.
  • Sun, W., Hu, Z. and Hong, L.J. (2018) Gaussian mixture model-based random search for continuous optimization via simulation in Proceedings of the 2018 Winter Simulation Conference, IEEE Press, Piscataway, NJ, pp. 2003–2014.
  • Tawarmalani, M. and Sahinidis, N.V. (2005) A polyhedral branch-and-cut approach to global optimization. Mathematical Programming, 103(2), 225–249.
  • van Ackooij, W. and Henrion, R. (2014) Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions. SIAM Journal on Optimization, 24(4), 1864–1889.
  • van Ackooij, W., Henrion, R., Möller, A. and Zorgati, R. (2010) On probabilistic constraints induced by rectangular sets and multivariate normal distributions. Mathematical Methods of Operations Research, 71(3), 535–549.
  • van Ackooij, W., Henrion, R., Möller, A. and Zorgati, R. (2014) Joint chance constrained programming for hydro reservoir management. Optimization and Engineering, 15(2), 509–531.
  • Wang, J. and Taaffe, M.R. (2015) Multivariate mixtures of normal distributions: Properties, random vector generation, fitting, and as models of market daily changes. INFORMS Journal on Computing, 27(2), 193–203.
  • Wilson, R. and Calway, A. (2001) Multiresolution Gaussian mixture models for visual motion estimation, in International Conference on Image Processing, IEEE, Thessaloniki, Greece, pp. 921–924.
  • Zeevi, A.J. and Meir, R. (1997) Density estimation through convex combinations of densities: Approximation and estimation bounds. Neural Networks, 10(1), 99–109.
  • Zymler, S., Kuhn, D. and Rustem, B. (2013) Distributionally robust joint chance constraints with second-order moment information. Mathematical Programming, 137(1-2), 167–198.

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