A Deep Generative Learning Approach for Two-stage Adaptive Robust Optimization

TL;DR

AGRO tackles high-dimensional uncertainty in two-stage adaptive robust optimization by learning a tight, data-driven uncertainty set with a variational autoencoder and performing adversarial generation via projected gradient ascent within a column-and-constraint generation framework. This approach replaces loose polyhedral or ellipsoidal sets with high-density regions in the learned latent space, enabling tighter probabilistic constraints without excessive conservatism. Across production-distribution and capacity expansion tests in power-system contexts, AGRO reduces total costs relative to standard CCG (up to 1.8% in production-distribution and up to 11.6% in capacity expansion) and yields more realistic worst-case contingencies, while improving scalability. The work provides a practical data-driven path to tractable, robust two-stage planning under high-dimensional uncertainty.

Abstract

Two-stage adaptive robust optimization (ARO) is a powerful approach for planning under uncertainty, balancing first-stage decisions with recourse decisions made after uncertainty is realized. To account for uncertainty, modelers typically define a simple uncertainty set over which potential outcomes are considered. However, classical methods for defining these sets unintentionally capture a wide range of unrealistic outcomes, resulting in overly-conservative and costly planning in anticipation of unlikely contingencies. In this work, we introduce AGRO, a solution algorithm that performs adversarial generation for two-stage adaptive robust optimization using a variational autoencoder. AGRO generates high-dimensional contingencies that are simultaneously adversarial and realistic, improving the robustness of first-stage decisions at a lower planning cost than standard methods. To ensure generated contingencies lie in high-density regions of the uncertainty distribution, AGRO defines a tight uncertainty set as the image of "latent" uncertainty sets under the VAE decoding transformation. Projected gradient ascent is then used to maximize recourse costs over the latent uncertainty sets by leveraging differentiable optimization methods. We demonstrate the cost-efficiency of AGRO by applying it to both a synthetic production-distribution problem and a real-world power system expansion setting. We show that AGRO outperforms the standard column-and-constraint algorithm by up to 1.8% in production-distribution planning and up to 11.6% in power system expansion.

Figures (5)

• Figure 1: The AGRO solution algorithm. (i) First-stage decisions ${\bm{x}}^*$ are obtained by solving a main problem, which approximates the original ARO uncertainty set $\mathcal{U}$ by a finite scenario set $\mathcal{S}$ (see problem \ref{['eq:MP']}) (ii) The latent variable ${\bm{z}}$ is sampled from within $\mathcal{Z}$ and decoded to obtain an initial ${\boldsymbol{\xi}}$. (iii) The recourse problem is solved given ${\bm{x}}^*$ and ${\boldsymbol{\xi}}$ and its optimal objective value is differentiated with respect to ${\boldsymbol{\xi}}$ and (after backpropagating through the decoder $f_\theta$) ${\bm{z}}$. (iv) A gradient ascent step is taken to update ${\bm{z}}$, which is then projected onto $\mathcal{Z}$ and decoded to obtain an updated ${\boldsymbol{\xi}}$. Steps (iii) and (iv) are iterated until converging to a worst-case ${\boldsymbol{\xi}}$. (v) The worst-case ${\boldsymbol{\xi}}$ is then added to $\mathcal{S}$, at which point the main problem is re-optimized.
• Figure 2: A comparison of classical uncertainty sets (a)-(c) with the proposed VAE-learned uncertainty set (d) for an illustrative bivariate distribution. By covering a smaller region in $\mathbb{R}^D$, the VAE-learned uncertainty set (d) is more likely to yield a tighter approximation of the probabilistic constraint in equation \ref{['eq:chance_constraint']} and less conservative first-stage decisions.
• Figure 3: (Left): Average cost improvement with standard deviations for AGRO over CCG with budget and ellipsoidal uncertainty sets. (Right): Box-plots showing relative improvement for AGRO over CCG with budget uncertainty. The relative cost advantage of VAE-learned uncertainty sets over classical uncertainty sets increases with the dimensionality of uncertainty.
• Figure 4: Comparison of worst-case uncertainty realizations as obtained by (a) CCG and (b) AGRO with $L=4$. Different colors are used to represent each of the six buses in the power system. A random sample of 100 observations is shown in gray to represent the historical distribution of load and capacity factors.
• Figure 5: Comparison of sample-estimated VaR, $\hat{F}^{-1}(\alpha; {\bm{x}}^*)$, and estimated worst-case costs obtained from solving the adversarial subproblem, $f({\boldsymbol{\xi}}^i, {\bm{x}}^*)$. Each point is obtained from one iteration of AGRO/CCG. Points that are closer to the diagonal line indicate a more accurate estimate of VaR. The middle and right-hand plots provide a zoomed-in view of two regions with a high concentration of points. Subproblem estimates are almost always greater than sample estimates except in the case of AGRO with $L=2$ (blue).