Scenario Reduction for Distributionally Robust Optimization

TL;DR

This work tackles computational challenges in distributionally robust optimization (DRO) by introducing a general scenario-reduction framework that projects the original ambiguity set onto a reduced set of representative scenarios. The method yields provable worst-case approximation guarantees via an $\alpha\beta$ bound, and is applicable to both discrete and continuous uncertainty distributions; it also extends to convex quadratic objectives through a mixed-integer semidefinite program. Key contributions include optimal clustering formulations (MIP and MISDP), partition-based bounds, and a dimension-reduction approach for interval and ellipsoidal ambiguity sets, with extensive experiments on MIPLIB instances and portfolio optimization that demonstrate substantial runtime reductions with modest loss in solution quality. The results show that fast heuristics like $k$-means can closely match optimal reductions in practice, while the proposed framework provides rigorous guarantees when needed. Overall, the paper delivers a scalable DRO reduction toolkit with theoretical bounds and practical validation across linear and quadratic objectives.

Abstract

Stochastic and (distributionally) robust optimization problems often become computationally challenging as the number of scenarios increases. Scenario reduction is therefore a key technique for improving tractability. We introduce a general scenario reduction method for distributionally robust optimization (DRO), which includes stochastic and robust optimization as special cases. Our approach constructs the reduced DRO problem by projecting the original ambiguity set onto a reduced set of scenarios. Under mild conditions, we establish bounds on the relative quality of the reduction. The methodology is applicable to random variables following either discrete or continuous probability distributions, with representative scenarios appropriately selected in both cases. Given the relevance of optimization problems with linear and quadratic objectives, we further refine our approach for these settings. Finally, we demonstrate its effectiveness through numerical experiments on mixed-integer benchmark instances from MIPLIB and portfolio optimization problems. Our results show that the proposed approximation significantly reduces solution time while maintaining high solution quality with only minor errors.

Figures (12)

• Figure 1: Approximation guarantee as a function of the representative scenario $\tilde{s} = (\tilde{s}_1, \tilde{s}_2)$ for scenario set $\mathcal{S}=[1,3]\times[1,2]\subset \mathbb{R}^2$.
• Figure 2: Variable substitution in proof of Lemma \ref{['lem:one_dim_approx_bound']} exemplarily for $K=4$.
• Figure 3: Covering of a scenario set (blue region) with hyperrectangles where $r_1 = 3$ and $r_2 = 2$ for $K = 6$.
• Figure 4: The mean approximation factor (left) and time factor (right) in dependence of the scenario reduction factor for $N=0$ samples for the two different clustering algorithms for the instance app2-2.
• Figure 5: The approximation guarantees (left) and the realized approximation factors (right) with the red line indicating the line through the origin for the instance app2-2. The coloring of the dots indicates the different values of $s_{inc}$ used in the creation of the scenarios. The cogs indicate the average AFs for the three values of $s_{inc}$.

Theorems & Definitions (32)

• Example 2.1
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Remark 1
• Lemma 3.1
• proof
• Corollary 3.2
• Theorem 3.3