Quantitative stability of fully random two-stage stochastic programs with mixed-integer recourse
- 10 Downloads
The quantitative stability and empirical approximation of two-stage stochastic programs with mixed-integer recourse are investigated. We first study the boundedness of optimal solutions to the second stage problem by utilizing relevant results for parametric (mixed-integer) linear programs. After that, we reestablish existing quantitative stability results with fixed recourse matrices by adopting different probability metrics. This helps us to extend our stability results to the fully random case under the boundedness assumption of integer variables. Finally, we consider the issue of empirical approximation, and the exponential rates of convergence of the optimal value function and the optimal solution set are derived when the support set is countable but may be unbounded.
KeywordsStochastic programming Mixed-integer recourse Quantitative stability Fully random Probability metric Exponential convergence
The authors are grateful to the relevant editors and two anonymous reviewers for their detailed and insightful comments and suggestions, which have led to a substantial improvement of the paper in both content and style.
- 1.Ahmed, S.: Two-stage stochastic integer programming: a brief introduction. Wiley Encyclopedia of Operations Research and Management Science. Wiley, Hoboken (2010)Google Scholar
- 2.Ahmed, S., Shapiro, A., Shapiro, E.: The sample average approximation method for stochastic programs with integer recourse. www.optimization-online.org (2002). Accessed 2 June 2002
- 9.Jiang, J., Chen, Z.: Quantitative stability of two-stage stochastic linear programs with full random recourse. www.optimization-online.org (2018). Accessed 5 Jan 2018
- 16.Römisch, W., Vigerske, S.: Recent progress in two-stage mixed-integer stochastic programming with applications to power production planning. In: Handbook of Power Systems I, pp. 177–208. Springer (2010)Google Scholar
- 22.Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on stochastic programming: modeling and theory, 2nd edn. MOS-SIAM Ser. Optim, SIAM, Philadelphia, PA (2009)Google Scholar
- 23.Zhao, C., Guan, Y.: Data-driven risk-averse two-stage stochastic program with \(\zeta \)-structure probability metrics. www.optimization-online.org (2015). Accessed 5 Nov 2015