H. Attouch and R. J.-B. Wets, “Quantitative Stability of Variational Systems: I. The Epigraphical Distance,” Tran- sactions of the American Mathematical Society, Vol. 328, No. 2, 1991, pp. 695-729. doi:10.2307/2001800

has been cited by the following article:

• TITLE: First Order Convergence Analysis for Sparse Grid Method in Stochastic Two-Stage Linear Optimization Problem

  AUTHORS: Shengyuan Chen

  KEYWORDS: Convergence Analysis, Stochastic Optimization, Scenario Generation, Convex Analysis, Measure Theory

  JOURNAL NAME: American Journal of Computational Mathematics, Vol.1 No.4, December 9, 2011

  ABSTRACT: Stochastic two-stage linear optimization is an important and widely used optimization model. Efficiency of numerical integration of the second stage value function is critical. However, the second stage value function is piecewise linear convex, which imposes challenges for applying the modern efficient spare grid method. In this paper, we prove the first order convergence rate of the sparse grid method for this important stochastic optimization model, utilizing convexity analysis and measure theory. The result is two-folded: it establishes a theoretical foundation for applying the sparse grid method in stochastic programming, and extends the convergence theory of sparse grid integration method to piecewise linear and convex functions.