On Feasibility of Sample Average Approximation Solutions

Abstract

When there are infinitely many scenarios, the current studies of two-stage stochastic programming problems rely on the relatively complete recourse assumption. However, such assumption can be unrealistic for many real-world problems. This motivates us to study the cases where the sample average approximation (SAA) solutions are not necessarily feasible. When the problems are convex and the true solutions lie in the interior of the set of feasible solutions, we show the portion of infeasible SAA solutions decays exponentially when the sample size increases. We also study functions with chain-constrained domain, and show the portion of SAA solutions having a low degree of feasibility decays exponentially when the sample size increases. This result is then extended to multistage stochastic programming.

On sample average approximation for two-stage stochastic programs without relatively complete recourse

“…Without this assumption, they demonstrate that in the chained-constrained case the dependence on the parameter m in the convergence rate can be replaced by the number of active constraints in an optimal solution, which, e.g., can be bounded by the number of first-stage variables n 1 . Our results complement those in [11] by conducting a different analysis which does not use the chain-constrained domain assumption nor an assumption that domF lies in the interior of X.…”

“…In this section we assume the set X is finite, such as in the case that all decision variables are integer and bounded. In [5], it has been shown that, in the case when f (x) is real-valued for all x ∈ X, under certain assumptions, the optimal solution set of ( 12) converges exponentially to the optimal solution set of (11). We extend this result to the extended-real-valued objective case.…”

“…Aside from a few classic results on convergence in the limit and expected bias of the SAA objective value, which we review in Section 2, the work that is most closely relate to our results is the recent paper by Liu [11], who establishes similar results on generating solutions with high recourse likelihood for stochastic programs having a property they refer to as chain-constrained domain. In their most general results, they establish that if the problem has a chained-constrained domain "of order m", then for ǫ ∈ (0, 1), the probability (over the sample) that the recourse likelihood of the SAA solution is less than 1 − ǫ decreases exponentially fast, provided the sample size is at least as large as m/ǫ.…”

“…We only need to prove that the constraints are equivalent in (11) and (18). A feasible solution of ( 18) is trivially a feasible solution of (11). For x ∈ X satisfying H(x, ξ) ≤ 0 for P-almost every ξ ∈ Ξ, let…”

On Distributionally Robust Multistage Convex Optimization: New Algorithms and Complexity Analysis

“…In [12], the ambiguity sets are taken to be finitely supported Wasserstein metric balls centered at the empirical distributions, and the algorithm is shown to converge asymptotically with stochastic sampling methods. We comment that all of the above variants of DDP algorithms rely on the assumption of relatively complete recourse, while it is indeed possible to have MSCO without such assumption [23].…”