Numerical Algebra, Control and Optimization (NACO)

Continuity and stability of two-stage stochastic programs with quadratic continuous recourse

Pages: 197 - 209, Volume 5, Issue 2, June 2015      doi:10.3934/naco.2015.5.197

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Zhiping Chen - Department of Computing Science, School of Mathematics and Statistics, Xi'an Jiaotong University, 710049 Xi'an, Shanxi, China (email)
Youpan Han - School of Science, Xi'an Polytechnic University,710048 Xi'an, Shanxi, China (email)

Abstract: For two-stage stochastic programs with quadratic continuous recourse where all the coefficients in the objective function and the right-hand side vector in the second-stage constraints vary simultaneously, we firstly show the locally Lipschtiz continuity of the optimal value function of the recourse problem, then under suitable probability metric, we derive the joint Lipschitz continuity of the expected optimal value function with respect to the first-stage variables and the probability distribution. Furthermore, we establish the qualitative and quantitative stability results of the optimal value function and the optimal solution set with respect to the Fortet-Mourier probability metric, when the underlying probability distribution is perturbed. Finally, we show the exponential convergence rate of the optimal value sequence when we solve two-stage quadratic stochastic programs by the sample average approximation method.

Keywords:   Stochastic quadratic programming, Lipschitz continuity, probability metric, stability, sample average approximation.
Mathematics Subject Classification:  Primary: 90C15, 90C20,90C31.

Received: September 2014;      Revised: May 2015;      Available Online: June 2015.