Subdifferential Convergence in Stochastic Programs
John R. Birge, Liqun Qi · SIAM Journal on Optimization · 1995
In this paper, we discuss convergence behavior of subdifferentials in approximation schemes for stochastic programs. This information is useful for solving stochastic programs by nonlinear programming techniques. Wets [Variational Inequalities and Complementarity Problems, John Wiley, New York, 1980, pp. 375–404] showed that epiconvergence of closed convex functions implies the set convergence of the graph of the subdifferentials of these functions. This conclusion is not true in general by a counterexample of Higle and Sen [Math. Oper. Res., 17 (1992), pp. 112–311]. We show that epiconvergence of closed convex functions implies set convergence of subdifferentials of these functions at points where the limit function is differentiable and apply this result to convex stochastic programs. We also show that similar results can be achieved in three other cases of expectational functionals: piecewise smooth integrands, continuous probability distributions, and loss functions. In the case of loss functions, we extend the existing results of Marti [Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 31 (1975), pp. 203–233] to more general situations. Some basic methods using the approximate derivative information are also discussed.