Convergence Analysis for Distributionally Robust Optimization and Equilibrium Problems

Published Online:https://doi.org/10.1287/moor.2015.0732

References

  • Aghassi M, Bertsimas D (2006) Robust game theory. Math. Programming 107:231–273.CrossrefGoogle Scholar
  • Athreya KB, Lahiri SN (2006) Measure Theory and Probability Theory (Springer, New York).Google Scholar
  • Attouch H, Buttazzo G, Michaille G (2005) Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization (SIAM, Philadelphia).Google Scholar
  • Bank B, Guddat J, Klatte D, Kummer D, Tammer K (1982) Nonlinear Parametric Optimization (Academic Verlag, Berlin).CrossrefGoogle Scholar
  • Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust Optimization (Princeton University Press, Princeton, NJ).CrossrefGoogle Scholar
  • Bertsimas D, Popescu I (2005) Optimal inequalities in probability theory: A convex optimization approach. SIAM J. Optim. 15:780–804.CrossrefGoogle Scholar
  • Bertsimas D, Doan XV, Natarajan K, Teo C-P (2010) Models for minimax stochastic linear optimization problems with risk aversion. Math. Oper. Res. 35:580–602.LinkGoogle Scholar
  • Billingsley P (1968) Convergence and Probability Measures (John Wiley & Sons, New York).Google Scholar
  • Bonnans JF, Shapiro A (2000) Perturbation Analysis of Optimization Problems (Springer, New York).CrossrefGoogle Scholar
  • Boyd S, Vandenberghe L (2004) Convex Optimization (Cambridge University Press, Cambridge, UK).CrossrefGoogle Scholar
  • Breton M, Hachem SE (1995) Algorithms for the solution of stochastic dynamic minimax problems. Computational Optim. Appl. 4:317–345.CrossrefGoogle Scholar
  • Breton M, Hachem SE (1995) A scenario aggregation algorithm for the solution of stochastic dynamic minimax problems. Stochastics and Stochastic Rep. 53:305–322.CrossrefGoogle Scholar
  • Delage E, Ye Y (2010) Distributionally robust optimization under moment uncertainty with application to data-driven problems. Oper. Res. 58(3):592–612.LinkGoogle Scholar
  • Dupačová J (1987) The minimax approach to stochastic programming and an illustrative application. Stochastics 20:73–88.CrossrefGoogle Scholar
  • Dupačová J (2011) Uncertanities in minimax stochastic programs. Optimization 60:10–11.CrossrefGoogle Scholar
  • Fan K (1953) Minimax theorems. Proc. National Acad. Sci. USA 39:42–47.CrossrefGoogle Scholar
  • Goh J, Sim M (2010) Distributionally robust optimization and its tractable approximations. Oper. Res. 58(4):902–917.LinkGoogle Scholar
  • Goldfarb D, Iyengar G (2003) Robust portfolio selection problems. Math. Oper. Res. 28(1):1–38.LinkGoogle Scholar
  • Hall JA, Brorsen BW, Irwin SH (1989) The distribution of futures prices: A test of stable Paretian and mixture of normals hypotheses. J. Financial Quant. Anal. 24:105–116.CrossrefGoogle Scholar
  • Hess C (1999) Conditional expectation and marginals of random sets. Pattern Recognition 32:1543–1567.CrossrefGoogle Scholar
  • Higham N (1988) Computing a nearest symmetric positive semidefinite matrix. Linear Algebra and Its Appl. 103:103–118.CrossrefGoogle Scholar
  • Klatte D (1987) A note on quantitative stability results in nonlinear optimization. Seminarbericht Nr. 90, Sektion Mathematik, Humboldt-Universität zu Berlin, Berlin, 77–86.Google Scholar
  • Liu Y, Xu H (2013) Stability and sensitivity analysis of stochastic programs with second order dominance constraints. Math. Programming 142:435–460.CrossrefGoogle Scholar
  • Peel D, McLachlan GJ (2000) Robust mixture modelling using t distribution. Statist. Comput. 10:339–348.CrossrefGoogle Scholar
  • Pflug GC (2003) Stochastic optimization and statistical inference. Rusczyński A, Shapiro A, eds. Stochastic Program, Handbooks in OR & MS, Vol. 10 (North-Holland, Amsterdam), 427–482.CrossrefGoogle Scholar
  • Pflug GC, Pichler A (2011) Approximations for probability distributions and stochastic optimization problems. Bertocchi M, Consigli G, Dempster MAH, eds. Stochastic Optimization Methods in Finance and Energy, Vol. 163 (Springer, Berlin), 343–387.CrossrefGoogle Scholar
  • Prokhorov YV (1956) Convergence of random processes and limit theorems in probability theory. Theory of Probabilty and Its Appl. 1:157–214.CrossrefGoogle Scholar
  • Qu SJ, Goh M (2012) Distributionally robust games with an application to supply chain. Working paper, Harbin Institute of Technology, Harbin, China.Google Scholar
  • Riis M, Andersen KA (2005) Applying the minimax criterion in stochastic recourse programs. Eur. J. Oper. Res. 165:569–584.CrossrefGoogle Scholar
  • Römisch W (2003) Stability of stochastic programming problems. Ruszczyński A, Shapiro A, eds. Stochastic Programming (Elsevier, Amsterdam), 483–554.CrossrefGoogle Scholar
  • Scarf H (1958) A min–max solution of an inventory problem. Arrow KS, Karlin S, Scarf HE, eds. Studies in the Mathematical Theory of Inventory and Production (Stanford University Press, Stanford, CA), 201–209.Google Scholar
  • So AMC (2011) Moment inequalities for sums of random matrices and their applications in optimization. Math. Programming 130: 125–151.CrossrefGoogle Scholar
  • Shapiro A (2001) On duality theory of conic linear problems. Goberna MA, López MA, eds. Semi-Infinite Programming: Recent Advances (Kluwer Academic Publishers, Dordrecht, the Netherlands), 135–165.CrossrefGoogle Scholar
  • Shapiro A (2003) Monte Carlo sampling methods. Rusczyński A, Shapiro A, eds. Stochastic Program, Handbooks in OR & MS, Vol. 10 (North-Holland, Amsterdam), 353–425.CrossrefGoogle Scholar
  • Shapiro A (2013) Consistency of sample estimates of risk averse stochastic programs. J. Appl. Probab. 50:533–541.CrossrefGoogle Scholar
  • Shapiro A, Ahmed S (2004) On a class of minimax stochastic programs. SIAM J. Optim. 14:1237–1249.CrossrefGoogle Scholar
  • Shapiro A, Kleywegt AJ (2002) Minimax analysis of stochastic problems. Optim. Methods Software 17:523–542.CrossrefGoogle Scholar
  • Shapiro A, Xu H (2008) Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation. Optimization 57:395–418.CrossrefGoogle Scholar
  • Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on Stochastic Programming: Modeling and Theory (SIAM, Philadelphia).CrossrefGoogle Scholar
  • Sun H, Xu H (2013) Asymptotic convergence analysis for distributional robust optimization and equilibrium problems. Optimization Online, http://www.optimization-online.org/DB_HTML/2013/05/3857.html.Google Scholar
  • Takriti S, Ahmed S (2002) Managing short-term electricity contracts under uncertainty: A minimax approach. ISyE Technical report, http://www2.isye.gatech.edu/∼sahmed/minmax.pdf.Google Scholar
  • Wang Z, Glynn PW, Ye Y (2014) Likelihood robust optimization for data-driven problems. Preprint, arXiv:1307.6279.Google Scholar
  • Wiesemann W, Kuhn D, Rustem B (2013) Robust Markov decision process. Math. Oper. Res. 38(1):153–183.LinkGoogle Scholar
  • Xu H, Caramanis C, Mannor S (2012) A distributional interpretation of robust optimization. Math. Oper. Res. 37(1):95–110.LinkGoogle Scholar
  • Žáčková J (1966) On minimax solution of stochastic linear programming problems. Časopis pro Pěstování Matematiky 91:423–430.Google Scholar
  • Zhu S, Fukushima M (2009) Worst-case conditional value-at-risk with application to robust portfolio management. Oper. Res. 57(5): 1155–1156.LinkGoogle Scholar
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