Abstract
The paper examines the effect of uncertainty on the solution of mathematical programming problems, using Bayesian techniques. We show that the statistical inference of the unknown parameter lies in the solution vector itself. Uncertainty in the data is modeled using sampling models induced by constraints. In this context, the objective is used as prior, and the posterior is efficiently applied via Monte Carlo methods. The proposed techniques provide a new benchmark for robust solutions that are designed without solving mathematical programming problems. We illustrate the benefits of a problem with known solutions and their properties, while discussing the empirical aspects in a real-world portfolio selection problem.
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References
Abdelaziz, F. B. (2012). Solution approaches for the multiobjective stochastic programming. European Journal of Operational Research, 216(1), 1–16. https://doi.org/10.1016/j.ejor.2011.03.033
Ben Daya, M. (1995). Line search techniques for the logarithmic barrier function in quadratic programming. Journal of the Operational Research Society, 46(3), 332–338. https://doi.org/10.2307/2584326
Ben Daya, M., & Al Sultan, K. S. (1997). A new penalty function algorithm for convex quadratic programming. European Journal of Operational Research, 101(1), 155–163. https://doi.org/10.1016/S0377-2217(96)00138-5
Ben-Tal, A., & Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769–805. https://doi.org/10.1287/moor.23.4.769
Ben-Tal, A., & Nemirovski, A. (1999). Robust solutions of uncertain linear programs. Operation Research Letters, 25(1), 1–13. https://doi.org/10.1016/S0167-6377(99)00016-4
Bertsimas, D., Pachamanova, D., & Sim, M. (2004). Robust linear optimization under general norms. Operations Research Letters, 32(6), 510–516. https://doi.org/10.1016/j.orl.2003.12.007
Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations Research, 52(1), 35–53. https://doi.org/10.1287/opre.1030.0065
Bonami, P., & Lejeune, M. A. (2009). An exact solution approach for portfolio optimization problems under stochastic and integer constraints. Operations Research, 57(3), 650–670. https://doi.org/10.1287/opre.1080.0599
Boukouvala, F., Misener, R., & Floudas, C. A. (2016). Global optimization advances in mixed-integer nonlinear programming, MINLP, and constrained derivative-free optimization. CDFO. European Journal of Operational Research, 252(3), 701–727. https://doi.org/10.1016/j.ejor.2015.12.018
Burer, S., & Vandenbussche, D. (2008). A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Mathematical Programming Series A, 113(2), 259–282. https://doi.org/10.1007/s10107-006-0080-6
Castro, J. (2009). A stochastic programming approach to cash management in banking. European Journal of Operational Research., 192(3), 963–974. https://doi.org/10.1016/j.ejor.2007.10.015
Charnes, A., & Cooper, W. W. (1963). Deterministic equivalents for optimizing and satisfying under chance constraints. Operations Research, 11(1), 18–39. https://doi.org/10.1287/opre.11.1.18
D’Ambrosio, C., & Lodi, A. (2013). Mixed integer nonlinear programming tools: An updated practical overview. Annals of Operations Research, 204(1), 301–320. https://doi.org/10.1007/s10479-012-1272-5
Dantzig, G. B., & Madansky, A. (1961). On the solution of two-stage linear programs under uncertainty; In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability vol. 1, University of California Press, pp. 165–176.
Darby-Dowman, K., Barker, S., Audsley, E., & Parsons, D. (2000). A two-stage stochastic programming with recourse model for determining robust planting plans in horticulture. Journal of the Operational Research Society, 51(1), 83–89. https://doi.org/10.1057/palgrave.jors.2600858
Demokan, N., & Land, A. H. (1981). A parametric quadratic program to solve a class of bicriteria decision problems. Journal of the Operational Research Society, 32(6), 477–488. https://doi.org/10.1057/jors.1981.98
Dumskis, V., & Sakalauskas, L. (2015). Nonlinear Stochastic programming involving CVaR in the objective and constraints. Informatica, 26(4), 569–591. https://doi.org/10.15388/Informatica.2015.65
El Ghaoui, L., & Lebret, H. (1997). Robust solutions to least-squares problems with uncertain data. SIAM Journal on Matrix Analysis and Applications., 18(4), 1035–1064. https://doi.org/10.1137/S0895479896298130
El Ghaoui, L., Oustry, F., & Lebret, H. (1999). Robust solutions to uncertain semidefinite programs. SIAM Journal on Optimization., 9(1), 33–52. https://doi.org/10.1137/S1052623496305717
Geweke, J. (1999). Using simulation methods for Bayesian econometric models: Inference, development and communication. Econometric Reviews., 18(1), 1–126. https://doi.org/10.1080/07474939908800428
Guigues, V., & Romisch, W. (2012). Sampling-based decomposition methods for multistage stochastic programs based on extended polyhedral risk measures. SIAM Journal on Optimization., 22(2), 286–312. https://doi.org/10.1137/100811696
Homem-de-Mello, T., & Pagnoncelli, B. K. (2016). Risk aversion in multistage stochastic programming: A modeling and algorithmic perspective. European Journal of Operational Research., 249(1), 188–199. https://doi.org/10.1016/j.ejor.2015.05.048
Jackson, M., & Staunton, M. D. (1999). Quadratic programming applications in finance using excel. Journal of the Operational Research Society., 50(12), 1256–1266. https://doi.org/10.1057/palgrave.jors.2600839
Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Management Science., 37(5), 519–531.
Korhonen, P., & Yu, G. Y. (1998). On computing objective function values in multiple objective quadratic-linear programming. European Journal of Operational Research., 106(1), 184–190. https://doi.org/10.1016/S0377-2217(98)00214-8
Kumbhakar, S. C., & Lovell, C. A. K. (2000). Stochastic frontier analysis. Cambridge University Press.
Mansini, R., Ogryczak, W., & Speranza, M. G. (2014). Twenty years of linear programming-based portfolio optimization. European Journal of Operational Research., 234(2), 518–535. https://doi.org/10.1016/j.ejor.2013.08.035
Mulvey, J. M., & Erkan, H. G. (2006). Applying CVaR for decentralized risk management of financial companies. Journal of Banking & Finance., 30(2), 627–644. https://doi.org/10.1016/j.jbankfin.2005.04.010
Philpott, A. B., & de Matos, V. L. (2012). Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion. European Journal of Operational Research., 218(2), 470–483. https://doi.org/10.1016/j.ejor.2011.10.056
Pichler, A., & Tomasgard, A. (2016). Nonlinear stochastic programming–with a case study in continuous switching. European Journal of Operational Research., 252(2), 487–501. https://doi.org/10.1016/j.ejor.2016.01.007
Powell, W. B. (2019). A unified framework for stochastic optimization. European Journal of Operational Research., 275(3), 795–821. https://doi.org/10.1016/j.ejor.2018.07.014
Sakalauskas, L. (2002). Nonlinear stochastic programming by Monte-Carlo estimators. European Journal of Operational Research., 137(3), 558–573. https://doi.org/10.1016/S0377-2217(01)00109-6
Sen, S., & Higle, J. L. (1999). An introductory tutorial on stochastic linear programming models. Interfaces, 29(2), 33–61. https://doi.org/10.1287/inte.29.2.33
Shapiro, A., Tekaya, W., da Costa, J. P., & Soares, M. P. (2013). Risk neutral and risk averse stochastic dual dynamic programming method. European Journal of Operational Research., 224(2), 375–391. https://doi.org/10.1016/j.ejor.2012.08.022
Soyster, A. (1973). Convex programming with set-inclusive constraints and applications to in-exact linear programming. Operation Research., 21(5), 1154–1157. https://doi.org/10.1287/opre.21.5.1154
Tierney, L. (1994). Markov chains for exploring posterior distributions. Annals of Statistics., 22(4), 1701–1728.
Tintner, G. (1955). Stochastic linear programming with applications to agricultural economics. In H. A. Antosiewicz (Ed.), Proceedings of the Second Symposium in Linear Programming (pp. 197–228), National Bureau of Standards, Washington, DC.
Tintner, G. (1960). A note on stochastic linear programming. Econometrica, 28(2), 490–495.
Uhan, N. A. (2015). Stochastic linear programming games with concave preferences. European Journal of Operational Research., 243(2), 637–646. https://doi.org/10.1016/j.ejor.2014.12.025
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All authors contributed to the study conception and design. Conceptualization, methodology, writing-original draft, editing and review, analysis, validation, and software were performed by Mike Tsionas, Dionisis Philippas and Constantin Zopounidis. The first draft of the manuscript was written by Mike Tsionas and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript and agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.
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Tsionas, M.G., Philippas, D. & Zopounidis, C. Exploring Uncertainty, Sensitivity and Robust Solutions in Mathematical Programming Through Bayesian Analysis. Comput Econ 62, 205–227 (2023). https://doi.org/10.1007/s10614-022-10277-z
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DOI: https://doi.org/10.1007/s10614-022-10277-z