Iterative Problem-Driven Scenario Reduction

• IPDSR is an advanced technique that uses decision-driven metrics to select representative scenarios in stochastic and robust optimization.
• Its iterative framework refines scenario selection based on performance metrics like optimality gap and tail risk to preserve decision quality.
• The method integrates problem-specific structures and rigorous formulations to achieve a reduced yet accurate scenario representation.

Iterative Problem-Driven Scenario Reduction (IPDSR) is an advanced methodology for compressing large sets of uncertainty scenarios in stochastic optimization, robust optimization, and optimal control, where representative scenarios are selected based on their direct impact on the optimization problem rather than on statistical proximity alone. IPDSR frameworks iteratively adapt the scenario selection according to the structure of the underlying problem and targeted performance metrics, such as optimality gap or tail-risk, achieving a tractable yet highly accurate reduced problem representation.

1. Conceptual Foundations

Conventional scenario reduction techniques focus on approximating a complex n-point empirical distribution by a simpler m-point distribution (m < n), often using distribution-driven distances like Wasserstein or energy metrics. In contrast, IPDSR integrates the downstream optimization problem's structure—objective sensitivities, feasibility under constraints, or risk measures—so the scenario selection is aligned with problem objectives and critical risk exposures.

Two broad reduction schemes are distinguished:

• Continuous Scenario Reduction: The reduced support points ("atoms") can be freely located in the scenario space (e.g., ℝᵈ), achieving a smaller approximation error due to higher flexibility.
• Discrete Scenario Reduction: Reduced scenarios are constrained to be drawn from the original set.

The approximation quality is typically measured using a distance metric such as the type-l Wasserstein distance:

$$d_l(\hat{P}_n,Q) = \left[ \min_{\Pi \in \mathbb{R}_+^{n \times m}} \sum_{i=1}^n \sum_{j \in J} \pi_{ij} \|\xi_i - \zeta_j\|^l \right]^{1/l}$$

IPDSR adapts this general paradigm by replacing the generic metric with a problem-driven metric (e.g., implementation error, decision applicability), or by integrating clustering directly in the decision space rather than in the outcome space.

2. Core Methodologies and Algorithms

A defining feature of IPDSR is the explicit embedding of the optimization problem's structure into the scenario reduction process via decision-aware clustering, selection, and feedback mechanisms.

Problem-Driven Clustering and Distance Functions: Instead of clustering based on the ℓ₁ or ℓ₂ distance between scenarios in distribution space, IPDSR frameworks employ a "Problem-Driven Distance" metric defined through opportunity cost or conditional objective value. For stochastic programming, for instance:

$$d(\xi_i, \zeta_k) = [F(z^*_{\zeta_k}, \xi_i) - F(z^*_{\xi_i}, \xi_i)] + [F(z^*_{\xi_i}, \zeta_k) - F(z^*_{\zeta_k}, \zeta_k)]$$

where $F(z^*_{\xi_i}, \cdot)$ is the cost of applying the optimal decision for scenario $\xi_i$ to another scenario.

Iterative Optimization and Scenario Update: The scenario reduction is performed within an iterative loop. In each iteration:

• The current representative scenarios are used to solve the reduced optimization problem.
• Ex-post performance (e.g., optimality gap, tail risk, implementation error) is evaluated against the full scenario set.
• New representative scenarios are selected, often by solving an MILP or MIP that minimizes the discrepancy between reduced and original objective values, explicitly preserving tail risks (such as CVaR) or worst-case costs.
• An aggregation step (e.g., cost vector aggregation) may be employed to ensure computational tractability.

Active and Strategic Scenario Selection: In probability distribution-free settings, the notion of "active scenarios"—scenarios that generate binding constraints at optimality—enables dramatic reduction in scenario set size. Strategic rules eliminate redundant (repeated) scenarios and select only those with distinct impact on the solution, often using a dissimilarity logic or machine learning-based selection heuristics.

3. Mathematical Formulations and Performance Guarantees

IPDSR frameworks provide rigorous mathematical formulations that align reduction with performance guarantees. Commonly, the selection of representative scenarios is rendered as a mixed-integer program:

• For stochastic optimization with CVaR:

$$\begin{align*} &\min_{\{v_{ij}, u_j\}}~ \left| \sum_{j=1}^{N} \sum_{i=1}^{N} v_{ij} \gamma_i (F_i - F_j + \frac{\lambda}{1-\alpha}([F_i - v_\xi^\alpha]_+ - [F_j - v_\zeta^\alpha]_+)) + \lambda (v_\xi^\alpha - v_\zeta^\alpha) \right| \ &\text{subject to}~v_{ij} \leq u_j,~v_{jj}=u_j,~\sum_j v_{ij}=1,~\sum_j u_j=K,~u_j,v_{ij} \in \{0,1\} \end{align*}$$

• For robust/combinatorial optimization, scenario reduction employs domination-based MILPs that ensure for all $x$ in feasible set $X$, the reduced scenario set represents the worst-case outcome within a given guarantee.

Strong a priori bounds exist for the impact of reduction. For Wasserstein reduction, for example, the worst-case type-2 distance for $n$ original and $m$ reduced scenarios on the unit ball satisfies:

$\bar{C}_2(n, m) \le \sqrt{\frac{n-m}{n-1}}$

Problem-driven frameworks may provide similar guarantees on the optimality gap or performance degradation as functions of the loss or spread induced by the representative set.

4. Evaluation Metrics and Validation Indices

Scenario reduction performance within IPDSR is assessed using ex-ante and ex-post indices, reflecting both statistical closeness and decision relevance:

• Optimality Gap (OG):

$$\mathrm{OG} = \frac{F(z^*_\zeta, \xi) - F(z^*_\xi, \xi)}{F(z^*_\xi, \xi)}$$

• Validation Similarity: Wasserstein distance between outcome distributions produced by solutions from reduced and original scenarios.
• Scenario Effectiveness (SE): Marginal increase in OG upon removing a particular representative scenario.
• Implementation Error: Difference in objective value when the reduced-problem solution is evaluated on all original scenarios.
• Problem-driven Davies-Bouldin Index: Measures within- and between-cluster compactness in the decision space.

5. Applications and Domain Integrations

IPDSR is applicable across diverse domains:

• Stochastic Programming: Reduces empirical scenario sets for large-scale stochastic problems (including power systems) while controlling performance loss and computational cost.
• Robust and Distributionally Robust Optimization: Delivers smaller, scenario sets for robust optimization (objective uncertainty) and DRO (with ambiguity sets), retaining tight worst-case guarantees (Aigner et al., 14 Mar 2025, Fairbrother et al., 11 Oct 2024).
• Risk-Averse Optimization: IPDSR is particularly potent in settings with tail risk (e.g., CVaR). The framework ensures that the representative scenario set maintains coverage of extreme events, supporting reliable, risk-averse dispatch and system design (Zhuang et al., 17 Oct 2025).
• Optimal Control: In semi-infinite or chance-constrained control, iterative scenario selection (often via worst-case scenario addition) maintains robust performance and constraint satisfaction with far fewer scenarios (Zagorowska et al., 2023, Cordiano et al., 11 Apr 2024).
• Power System Operation: Domain-embedded scenario reduction enables optimal scheduling and economic dispatch with bounded deviation (e.g., <0.1% OG (Zhuang et al., 11 Apr 2024)) from full-size stochastic models.

6. Comparative Analysis and Limitations

IPDSR stands in contrast to classic, distribution-driven scenario reduction (DDSR), which typically employs statistical metrics (e.g., Euclidean, Wasserstein, energy distance) and often ignores the effect of scenario selection on optimization-relevant features—potentially omitting critical tail events or introducing large optimality gaps.

A notable finding is that simple heuristics or greedy selection algorithms, such as k-means or classical greedy reduction, can yield arbitrarily poor solutions in worst-case instances, lacking guarantees on worst-case performance or risk-coverage (Rujeerapaiboon et al., 2017). In domains with complex nonconvexities or endogenous uncertainty, iterative PD-driven selection (including closed-loop learning or local reduction methods) yields higher robustness and computational efficiency (Li, 2021, Zagorowska et al., 2022, Zagorowska et al., 2023).

A limitation is that problem-driven (especially iterative, MIP-based) reduction may be computationally intensive in very large-scale settings. Aggregation strategies, careful formulation, or parallelism are often required for tractability. Moreover, the effectiveness depends on the fidelity of the problem-driven metric to decision value.

7. Future Directions and Open Challenges

Recent developments indicate several promising avenues:

• Integration of Advanced Risk Measures: Extension toward other risk-sensitive criteria (e.g., spectral risk measures, distributionally robust CVaR), building upon the CVaR-focused IPDSR frameworks (Zhuang et al., 17 Oct 2025).
• Learning-Based Selection: Machine-learning heuristics for scenario relevance prediction offer further reduction by exploiting scenario and problem features, ranking scenario importance, and guiding selection (Goerigk et al., 2022).
• Iterative Feedback Loops: Embedding scenario reduction within online optimization and adaptively updating the representative set as further data or model feedback becomes available.
• Mathematical Tightening of Guarantees: Development of sharper theoretical bounds for more complex or nonlinear objectives, as well as for continuous scenario spaces.
• Scalability and Real-Time Implementation: Aggregation, decomposition, and parallel computing strategies to maintain the tractability of MIP-based or clustering-based PD-driven reductions in operational settings.

The evolving framework of IPDSR synthesizes advances in scenario reduction, optimization, risk analytics, and machine learning, yielding flexible yet principled methods for compressing uncertainty in large-scale and risk-sensitive decision models.