Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Latest Articles
44
Views
0
CrossRef citations to date
0
Altmetric
Research Article

On regularization schemes for data-driven optimization based on Cressie–Read divergence and CVaR

ORCID Icon
Received 15 Sep 2022, Accepted 12 Apr 2024, Published online: 29 Apr 2024

References

  • Ben-Tal A, Ghaoui LE, Nemirovski A. Robust optimization. Princeton: Princeton University Press; 2009.
  • Ben-Tal A, Nemirovski A. Robust solutions to uncertain programs. Oper Res Lett. 1999;25(1):1–13. doi: 10.1016/S0167-6377(99)00016-4
  • El-Ghaoui L, Oustry F, Lebret H. Robust solutions to uncertain semidedefinite programs. SIAM J Optim. 1998;9(1):33–52. doi: 10.1137/S1052623496305717
  • Bertsimas D, Brown DB, Caramanis C. Theory and applications of robust optimization. SIAM Review. 2011;53(3):464–501. doi: 10.1137/080734510
  • Birge J, Louveaux F. Introduction to stochastic programming. New York: Springer Verlag; 2011.
  • Shapiro A, Dentcheva D, Ruszczynski A. Lectures on stochastic programming: modeling and theory. 2nd ed. Philadelphia: SIAM; 2009.
  • Ben-Tal A, Nemirovski A. Robust solutions of linear programming problems contaminated with uncertain data. Math Program. 2000;88(3):411–424. doi: 10.1007/PL00011380
  • Klabjan D, Simchi-Levi D, Song M. Robust stochastic lot-sizing by means of histograms. Prod Oper Manag. 2013;22(3):691–710. doi: 10.1111/j.1937-5956.2012.01420.x
  • Bertsimas D, Thiele A. A data-driven approach to newsvendor problems. Technical report, Massachusetts Institute of Technology. 2004.
  • Bertsimas D, Gupta V, Kallus N. Data-driven robust optimization. Math Program. 2018;167(2):235–292. doi: 10.1007/s10107-017-1125-8
  • Smith JE, Winkler RL. The optimizer's curse: skepticism and postdecision surprise in decision analysis. Manage Sci. 2006;52(3):311–322. doi: 10.1287/mnsc.1050.0451
  • Scarf HE. A min-max solution of an inventory problem. Technical report, RAND CORP SANTA MONICA CALIF. 1957.
  • Sinha A, Namkoong H, Duchi J. Certifying some distributional robustness with principled adversarial training. In: International Conference on Learning Representations; 2018.
  • Staib M, Jegelka S. Distributionally robust optimization and generalization in kernel methods. Adv Neural Inf Process Syst. 2019;32:9134–9144.
  • Gao R, Chen X, Kleywegt AJ. Distributional robustness and regularization in statistical learning. arXiv:171206050. 2017. p. 1–27.
  • Shaéezadeh-Abadeh S, Kuhn D, Mohajerin Esfahani P. Regularization via mass transportation. J Mach Learn Res. 2019;20(103):1–68.
  • Blanchet J, Kang Y, Murthy K. Robust wasserstein profile inference and applications to machine learning. J Appl Probab. 2019;56(3):830–857. doi: 10.1017/jpr.2019.49
  • Esfahani PM, Kuhn D. Data-driven distributionally robust optimization using the wasserstein metric: performance guarantees and tractable reformulations. Math Program. 2018;171(1-2):115–166. doi: 10.1007/s10107-017-1172-1
  • Gao R, Kleywetg AJ. Distributionally robust stochastic optimization with wasserstein distance. Math Oper Res. 2022;48(2):603–655. doi: 10.1287/moor.2022.1275
  • Ben-Tal A. Robust solutions of optimization problems affected by uncertain probabilities. Manage Sci. 2013;59(2):341–357. doi: 10.1287/mnsc.1120.1641
  • Duchi JC, Glynn PW, Namkoong H. Statistics of robust optimization: a generalized empirical likelihood approach. Math Oper Res. 2021;46(3):946–969. doi: 10.1287/moor.2020.1085
  • Duchi JC, Namkoong H. Variance-based regularization with convex objectives. J Mach Learn Res. 2018;19:1–55.
  • Bayraksan G, Love DK. Data-driven stochastic programming using phi-divergence. Oper Res Revol. 2015;1–19.
  • Gotoh J, Kim MJ, Lim AEB. Robust empirical optimization is almost the same as mean-variance optimization. Oper Res Lett. 2018;46(4):448–452. doi: 10.1016/j.orl.2018.05.005
  • Hu Z, Hong JL. Kullback–Leibler divergence constrained distributionally robust optimization. Available at Optimization Online: 2013. https://hdl.handle.net/1783.1/70889:1695–1724.
  • Namkoong H, Duchi JC. Variance-based regularization with convex objectives. Adv Neural Inf Process Syst. 2017;30.
  • Delage E, Ye Y. Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper Res. 2010;58(3):595–612. doi: 10.1287/opre.1090.0741
  • Hanasusanto GA, Roitch V, Kuhn D, et al. A distributionally robust perspective on uncertainty quantification and chance constrained programming. Math Program Ser B. 2015;151(1):35–62. doi: 10.1007/s10107-015-0896-z
  • Rahimian H, Mehrotra S. Distributionally robust optimization: A review. Arxiv preprint, arXiv:190805659. 2019.
  • Fournier N, Guillin A. On the rate of convergence in Wasserstein distance of the empirical measure. Probab Theory Relat Fields. 2015;162(3-4):707–738. doi: 10.1007/s00440-014-0583-7
  • Shafieezadeh-Abadeh S, Esfahani PM, Kuhn D. Distributionally robust logistic regression. Adv Neural Inform Process Syst. 2015;1:1576–1584.
  • Zhao C, Guan Y. Data-driven risk-averse stochastic optimization with wasserstein metric. Oper Res Lett. 2018;46(2):262–267. doi: 10.1016/j.orl.2018.01.011
  • Lam H. Robust sensitivity analysis for stochastic systems. Math Oper Res. 2016;41(4):1248–1275. doi: 10.1287/moor.2015.0776
  • Sun H, Xu H. Distributionally robust optimization and equilibrium problems. Math Oper Res. 2016;41(2):377–401. doi: 10.1287/moor.2015.0732
  • Maurer A, Pontil M. Empirical Bernstein bounds and sample variance penalization. Stat. 2009;1050:21.
  • Wu D, Zhu H, Zhou E. A Bayesian risk approach to data-driven stochastic optimization: formulations and asymptotics. SIAM J Optim. 2018;28(2):1588–1612. doi: 10.1137/16M1101933
  • Ni W, Jiang ZP. On regularization schemes for data-driven optimization. In: 2019 Chinese Control And Decision Conference (CCDC); IEEE; 2019. p. 3016–3023.
  • Rockafellar RT, Uryasev S. Optimization of conditional value-at-risk. J Risk. 2000;2(3):21–41. doi: 10.21314/JOR.2000.038
  • Uğurlu K. A new coherent multivariate average-value-at-risk. Optimization. 2021;72(2):493–519. doi: 10.1080/02331934.2021.1970755:1–27
  • Uğurlu K. Refinements of kusuoka representations on L∞. Optimization. 2022;71(11):3351–3362. doi: 10.1080/02331934.2022.2038152
  • Friedman JH. Multivariate adaptive regression splines. Ann Stat. 1991;19(1):1–67.
  • Weber GW, Batmaz I, Köksal G, et al. CMARS: a new contribution to nonparametric regression with multivariate adaptive regression splines supported by continuous optimization. Inverse Probl Sci Eng. 2012;20(3):371–400. doi: 10.1080/17415977.2011.624770
  • Özmen A, Weber GW. RMARS: robustification of multivariate adaptive regression spline under polyhedral uncertainty. J Comput Appl Math. 2014;259:914–924. doi: 10.1016/j.cam.2013.09.055
  • Özmen A, Weber GW, Batmaz İ, et al. RCMARS: robustification of CMARS with different scenarios under polyhedral uncertainty set. Commun Nonlinear Sci Numer Simul. 2011;16(12):4780–4787. doi: 10.1016/j.cnsns.2011.04.001
  • Cressie N, Read TR. Multinomial goodness-of-fit tests. J R Stat Soc: Ser B (Methodol). 1984;46(3):440–464. doi: 10.1111/j.2517-6161.1984.tb01318.x
  • Baggerly KA. Empirical likelihood as a goodness-of-fit measure. Biometrika. 1998;85(3):535–547. doi: 10.1093/biomet/85.3.535
  • Breuer T, Csiszár I. Systematic stress tests with entropic plausibility constraints. J Bank Financ. 2013;37(5):1552–1559. doi: 10.1016/j.jbankfin.2012.04.013
  • Nguyen X, Wainwright MJ, Jordan MI. Estimating divergence functionals and the likelihood ratio by convex risk minimization. IEEE Trans Inform Theory. 2010;56(11):5847–5861. doi: 10.1109/TIT.2010.2068870
  • Ahmadi-Javid A. Entropic value-at-risk: a new coherent risk measure. J Optim Theory Appl. 2012;153(3):1105–1123. doi: 10.1007/s10957-011-9968-2
  • Chouzenoux E, Gérard H, Pesquet JC. General risk measures for robust machine learning. arXiv preprint arXiv:190411707. 2019.
  • Birrell J, Katsoulakis MA, Pantazis Y. Optimizing variational representations of divergences and accelerating their statistical estimation. IEEE Trans Inform Theory. 2022;68(7):4553–4572. doi: 10.1109/TIT.2022.3160659
  • Csiszár I. The method of types. IEEE Trans Inform Theory. 1998;44(6):2505–2523. doi: 10.1109/18.720546
  • Owen AB. Empirical likelihood ratio condifence regions. Ann Stat. 1990;18(1):90–120. doi: 10.1214/aos/1176347494
  • Lam H, Zhou E. Quantifying uncertainty in sample average approximation. In: 2015 Winter Simulation Conference (WSC); IEEE; 2015. p. 3846–3857.
  • Lam H. Recovering best statistical guarantees via the empirical divergence-based distributionally robust optimization. Oper Res. 2019;67(4):1090–1105.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.