A comment on “computational complexity of stochastic programming problems”

Abstract

Although stochastic programming problems were always believed to be computationally challenging, this perception has only recently received a theoretical justification by the seminal work of Dyer and Stougie (Mathematical Programming A, 106(3): [423][424][425][426][427][428][429][430][431][432] 2006). Amongst others, that paper argues that linear two-stage stochastic programs with fixed recourse are #P-hard even if the random problem data is governed by independent uniform distributions. We show that Dyer and Stougie's proof is not correct, and we offer a correction which establishes the stronger result that even the approximate solution of such problems is #P-hard for a sufficiently high accuracy. We also provide new results which indicate that linear two-stage stochastic programs with random recourse seem even more challenging to solve.

Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations

“…It seems that there is no simple exact reformulation of (24) for arbitrary convex uncertainty sets . Interchanging the maximization over θ with the minimization over z in (24) would lead to the conservative upper bound of Corollary 4.3. Here, however, we employ an alternative approximation.…”

“…This phenomenon is termed the optimizer's curse and is reminiscent of overfitting effects in statistics [48]. Second, in order to evaluate the objective function of a stochastic program for a fixed decision x, we need to compute a multivariate integral, which is #P-hard even if h(x, ξ) constitutes the positive part of an affine function, while ξ is uniformly distributed on the unit hypercube [24,Corollary 1].…”

Regularized and Distributionally Robust Data-Enabled Predictive Control

“…Hence, we are unable to predict future output trajectories y, and we are missing necessary information to compute the expectation. Note that even in the case when system (1) and P v are known, solving (2) would require high-dimensional integration and is often computationally intractable [17]. One could also consider including joint output chance constraints in the problem setup (2) which would pose similar challenges as above.…”

Two-stage linear decision rules for multi-stage stochastic programming

“…. We observe that the subproblem (27) can be further decomposed into two: one problem including only the u + and u − variables, and the other problem including the remaining set of variables.…”

“…A significant challenge to using two-stage LDRs is that the resulting 2SLP is in general intractable to solve exactly. Indeed, 2SLP is #P -hard [17,27] due to the difficulty in evaluating the expectation of the recourse function. However, as argued in [57], under mild conditions Monte Carlo sampling-based methods can provide solutions of modest accuracy to a 2SLP (such a statement cannot be made for MSLP).…”