An Iterative Problem-Driven Scenario Reduction Framework for Stochastic Optimization with Conditional Value-at-Risk

TL;DR

This work tackles the computational burden of tail-risk aware SBSO-CVaR by introducing IPDSR, an iterative, problem-driven scenario reduction framework that projects the distribution space onto the SBSO-CVaR problem space and refines representative scenarios with a tractable MIP. It minimizes the optimality gap $OG = F(z^*_{\bm{\zeta}},\bm{\xi}) - F(z^*_{\bm{\xi}},\bm{\xi})$ while preserving tail risk, aided by objective aggregation and an explicit projection $\{F_i = f(z^*_{\bm{\zeta}},\xi_i)\}$ that informs clustering around decision-relevant features. The framework introduces ex-post evaluation indices—validation distribution similarity, OG, and representative scenario effectiveness—to quantify SR quality, and demonstrates superior performance on a day-ahead risk-averse VPP offering problem, achieving an OG below 1% with substantially reduced scenario sets and manageable computation times. By embedding problem structure into SR and providing a scalable MIP-based solution, IPDSR offers more reliable tail-risk representation and decision quality than traditional distribution-driven SR methods. The approach holds promise for broader adoption in risk-averse stochastic optimization where tail events critically influence operational decisions.

Abstract

Scenario reduction (SR) alleviates the computational complexity of scenario-based stochastic optimization with conditional value-at-risk (SBSO-CVaR) by identifying representative scenarios to depict the underlying uncertainty and tail risks. Existing distribution-driven SR methods emphasize statistical similarity but often exclude extreme scenarios, leading to weak tail-risk awareness and insufficient problem-specific representativeness. Instead, this paper proposes an iterative problem-driven scenario reduction framework. Specifically, we integrate the SBSO-CVaR problem structure into SR process and project the original scenario set from the distribution space onto the problem space. Subsequently, to minimize the SR optimality gap with acceptable computation complexity, we propose a tractable iterative problem-driven scenario reduction (IPDSR) method that selects representative scenarios that best approximate the optimality distribution of the original scenario set while preserving tail risks. Furthermore, the iteration process is rendered as a mixed-integer program to enable scenario partitioning and representative scenarios selection. And ex-post problem-driven evaluation indices are proposed to evaluate the SR performance. Numerical experiments show IPDSR significantly outperforms existing SR methods by achieving an optimality gap of less than 1% within an acceptable computation time.

Figures (4)

• Figure 1: Illustration example of non-linear mapping between distribution space and problem space: (a) scenarios in the problem space; (b) four example scenarios in the distribution space: left – net load scenarios; right – corresponding price scenarios.
• Figure 2: Illustration example of the iterative scenario reduction process: (a) initial values of $\{f(z^\ast_{\bm{\zeta}},\xi_i)\}_{i=1}^N$; (b) aggregated values of $\{f(z^\ast_{\bm{\zeta}},\xi_i)\}_{i=1}^N$; (c) reduced values of $\bm{\zeta}$ after one iterative process.
• Figure 3: The iterative curve of the OG value of IPDSR.
• Figure 4: Comparing results for different $N$ and $K$: (a) $N=400$, (b) $N=1000$.