Augmented Markov Chain Monte Carlo Simulation for Two-Stage Stochastic Programs with Recourse

References

• Aarts E, Korst J (1988) Simulated Annealing and Boltzmann Machines: A Stochastic Approach to Combinatorial Optimization and Neural Computing (John Wiley & Sons, New York).Google Scholar
• Bayraksan G, Morton DP (2009) Assessing solution quality in stochastic programs via sampling. Oskoorouchi MR, ed. Tutorials in Operations Research (INFORMS, Hanover, MD), 102–122.Link, Google Scholar
• Benders JF (1962) Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4:238–252.Crossref, Google Scholar
• Bielza C, Müller P, Insua DR (1999) Decision analysis by augmented probability simulation. Management Sci. 45(7):995–1007.Link, Google Scholar
• Birge JR, Louveaux FV (1988) A multicut algorithm for two stage stochastic linear programs. Eur. J. Oper. Res. 34(3):384–392.Crossref, Google Scholar
• Birge JR, Louveaux F (2011) Introduction to Stochastic Programming (Springer-Verlag, Berlin).Crossref, Google Scholar
• Brooks SP, Roberts GO (1998) Assessing convergence of Markov chain Monte Carlo algorithms. Statist. Comput. 8(4):319–335.Crossref, Google Scholar
• Dai L, Chen CH, Birge JR (2000) Convergence properties of two-stage stochastic programming. J. Optim. Theory Appl. 106(3):489–509.Crossref, Google Scholar
• Dantzig GB (1955) Linear programming under uncertainty. Management Sci. 1(3–4):197–206.Link, Google Scholar
• Dantzig GB, Thapa MN (1997) Linear Programming 2: Theory and Extensions (Springer-Verlag, New York).Google Scholar
• Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 339(1):1–38.Google Scholar
• Ermoliev Y (1988) Stochastic quasigradient methods. Ermoliev Y, Wets RJ-B, eds. Numerical Techniques for Stochastic Optimization (Springer-Verlag, Berlin), 141–185.Crossref, Google Scholar
• Gaivoronski A (1988) Stochastic quasigradient methods and their implementation. Ermoliev Y, Wets RJ-B, eds. Numerical Techniques for Stochastic Optimization (Springer-Verlag, Berlin), 313–351.Crossref, Google Scholar
• Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Statist. Sci. 7(4):457–472.Crossref, Google Scholar
• Geman D, Geman SA (1987) Relaxation and annealing with constraints. Complex Systems Technical Report 35, Division of Applied Mathematics, Brown University, Providence, RI.Google Scholar
• Higle JL, Sen S (1991) Stochastic decomposition: An algorithm for two-stage linear programs with recourse. Math. Oper. Res. 16(3):650–669.Link, Google Scholar
• Higle JL, Sen S (1996) Stochastic Decomposition: A Statistical Method for Large Scale Stochastic Linear Programming (Kluwer Academic Publishers, Norwell, MA).Crossref, Google Scholar
• Homem-de-Mello T, Bayraksan G (2014) Monte Carlo sampling-based methods for stochastic optimization. Working paper, Universidad Adolfo Ibanez, Santiago, Chile. http://www.optimization-online.org/DB_FILE/2013/06/3920.pdf.Crossref, Google Scholar
• Infanger G (1993) Monte Carlo (importance) sampling within a Benders decomposition algorithm for stochastic linear programs. Ann. Oper. Res. 39(1):69–95.Crossref, Google Scholar
• Isaacson DL, Madsen RW (1976) Markov Chains, Theory and Applications, Vol. 4 (John Wiley & Sons, New York).Google Scholar
• Jacquier E, Johannes M, Polson N (2007) MCMC maximum likelihood for latent state models. J. Econometrics 137(2):615–640.Crossref, Google Scholar
• Jacquier E, Johannes M, Polson N (2010) Maximum expected utility via MCMC. Working paper, Centre for Interuniversity Research and Analysis on Organizations, HEC Montréal, Montréal. http://faculty.chicagobooth.edu/nicholas.polson/research/papers/mcmceu.pdf.Google Scholar
• Jonsbråten TW, Wets RJ-B, Woodruff DL (1998) A class of stochastic programs with decision dependent random elements. Ann. Oper. Res. 82:83–106.Crossref, Google Scholar
• Kirkpatrick S, Gelatt CD Jr, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680.Crossref, Google Scholar
• Mak WK, Morton DP, Wood RK (1999) Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24(1–2):47–56.Crossref, Google Scholar
• Müller P (1999) Simulation-based optimal design. Bernardo JM, Berger JO, Dawid AP, Smith AFM, eds. Bayesian Statistics 6 (Oxford University Press, Oxford, UK), 459–474.Google Scholar
• Müller P, Sansó B, De Iorio M (2004) Optimal Bayesian design by inhomogeneous Markov chain simulation. J. Amer. Statist. Assoc. 99(467):788–798.Crossref, Google Scholar
• Parpas P, Ustun B, Webster M, Tran QK (2013) Importance sampling in stochastic programming: A Markov chain Monte Carlo approach. Working paper, Imperial College London, London. http://www.optimization-online.org/DB_FILE/2012/08/3558.pdf.Google Scholar
• Pennanen T, Koivu M (2005) Epi-convergent discretizations of stochastic programs via integration quadratures. Numer. Math. 100(1):141–163.Crossref, Google Scholar
• Pincus M (1968) A closed form solution of certain programming problems. Oper. Res. 16(3):690–694.Link, Google Scholar
• Pincus M (1970) A Monte Carlo method for the approximate solution of certain types of constrained optimization problems. Oper. Res. 18(6):1225–1228.Link, Google Scholar
• Polson NG, Sorensen M (2011) A simulation-based approach to stochastic dynamic programming. Appl. Stochastic Models Bus. Indust. 27(2):151–163.Crossref, Google Scholar
• Rubinstein RY, Shapiro A (1993) Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method, Vol. 346 (John Wiley & Sons, New York).Google Scholar
• Shapiro A (2003) Monte Carlo sampling methods. Ruszczyński A, Shapiro A, eds. Stochastic Programming, Handbooks in Operations Research and Management Science, Vol. 10 (Elsevier, Amsterdam), 353–425.Crossref, Google Scholar
• Shapiro A, Homem-de-Mello T (1998) A simulation-based approach to two-stage stochastic programming with recourse. Math. Programming 81(3):301–325.Crossref, Google Scholar
• Shapiro A, Homem-de-Mello T (2000) On the rate of convergence of optimal solutions of Monte Carlo approximations of stochastic programs. SIAM J. Optim. 11(1):70–86.Crossref, Google Scholar
• Shapiro A, Dentcheva D, Ruszczyński AP (2009) Lectures on Stochastic Programming: Modeling and Theory (Society for Industrial and Applied Mathematics, Philadelphia).Crossref, Google Scholar
• Spall JC (2005) Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control (John Wiley & Sons, Hoboken, NJ).Crossref, Google Scholar
• Tierney L (1994) Markov chains for exploring posterior distributions. Ann. Statist. 22(4):1701–1728.Crossref, Google Scholar
• van Laarhoven PJ, Aarts EHL (1987) Simulated Annealing: Theory and Applications (Reidel, Amsterdam).Crossref, Google Scholar
• Van Slyke RM, Wets R (1969) L-shaped linear programming with application to optimal control and stochastic programming. SIAM J. Appl. Math. 17(4):638–663.Crossref, Google Scholar