Data-driven two-stage conic optimization with zero-one uncertainties

Abstract

We address high-dimensional zero-one random parameters in two-stage convex conic optimization problems. Such parameters typically represent failures of network elements and constitute rare, high-impact random events in several applications. Given a sparse training dataset of the parameters, we motivate and study a distributionally robust formulation of the problem using a Wasserstein ambiguity set centered at the empirical distribution. We present a simple, tractable, and conservative approximation of this problem that can be efficiently computed and iteratively improved. Our method relies on a reformulation that optimizes over the convex hull of a mixed-integer conic programming representable set, followed by an approximation of this convex hull using lift-and-project techniques. We illustrate the practical viability and strong out-of-sample performance of our method on nonlinear optimal power flow problems affected by random contingencies, and we report improvements of up to 20% over existing methods. MotivationThis work is motivated by optimization problems arising in applications that are affected by an extremely large, yet finite, number of rare, high-impact random events. In particular, we are motivated by applications in which the decision-relevant random events consist of high-dimensional binary outcomes. Such applications are ubiquitous in network optimization, where the uncertain 1

A Lagrangian Dual Method for Two-Stage Robust Optimization with Binary Uncertainties

“…In the presence of binary uncertainties, the two-stage robust optimization problem P is NP-hard even if there are no first-stage decisions (i.e., X is a singleton) and the second-stage problem is a two-dimensional linear program with only uncertain objective coefficients (i.e., Y = R 2 + and h(ξ) is deterministic) [51]. Indeed, it is solvable in polynomial time only in few special cases, such as when the uncertainty set Ξ has a small inner description (i.e., in terms of a polynomial number of extreme points) or when h(ξ) = 0 and the matrices describing the constraints of Ξ and the slopes of the affine function d(ξ) are totally unimodular [43,51].…”

“…Several of these methods, particularly ones that are based on K-adaptability [27,49] and convexification [43,34,51], have also been extended to problems with mixed-integer uncertainties.…”

“…Before we proceed to discuss the Benders decomposition and column-and-constraint generation algorithms in some detail, we note that alternative approximation schemes for P and P I can be devised by first ignoring the binary restriction on the uncertain parameters, and then using any (exact or approximate) method that is designed to address problems with continuous uncertainties. However, recent studies [51,43] have shown that such approximations result in unnecessarily conservative (i.e., suboptimal) solutions even when the first-and second-stage feasible regions X and Y are convex.…”