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

Authors: Aron Brenner, Rahman Khorramfar, Jennifer Z Sun, Saurabh Amin
Primary area: optimization
Keywords: robust optimization, stochastic optimization, discrete optimization, deep learning, unsupervised learning

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 8% in power system expansion.

Decision: accept (poster)
Reviewer scores: 5, 6, 6, 6

Reviews and discussion:
<thread>
creation date:
1730044838815

thread_type:
official_review

<thread_object>
creator: reviewer_gwvx

<content>
summary:
This paper proposes the AGRO algorithm, which performs adversarial generation for two-stage adaptive robust optimization using a variational autoencoder. By decomposing the optimization problem to solving the 'main' problem and the adversarial subproblem iteratively, AGRO can provide tighter uncertainty estimation and lead to better optimization outputs.

strengths:
1. The paper is well-written, easy to follow, and understand.

2. With VAE-learned uncertainty, the proposed AGRO method does tighten the uncertainty bounds and leads to better optimization outcomes. An intuitive example in Figure 2 and experimental results clearly demonstrate this.

weaknesses:
1. In Section 3.2, the author proposes a projected gradient ascent heuristic method to optimize $q$. Although this PGA method is well-explained in the article and I understand why the author uses it, I still expect an ablation study on directly optimizing $q$ to show if PGA could still guarantee some level of optimization quality and if there are any speed improvements.

2. Although the proposed AGRO method is an improvement based on the CCG method, in the experiments section, the author should compare with more baselines for the two-step optimization problem, which I believe is a well-studied problem with many methods proposed to solve it.

questions:
My main question for this work is: Is the optimization problem the author wants to solve exactly a linear optimization problem (see Eq. 1, 7, 9, 10)? If so, there are already many tools for solving linear optimization problems, so why is the proposed method better than those?

If not, what kind of optimization problem does AGRO solve? Only convex?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730772669346

thread_type:
official_review

<thread_object>
creator: reviewer_a31w

<content>
summary:
This paper addresses the two-stage adaptive robust optimization (ARO) problem, where a key challenge is constructing an effective uncertainty set. The authors propose using a deep generative model to learn the uncertainty set, aiming to avoid overly conservative optimization. The method is evaluated on a synthetic production-distribution problem and a regional power system expansion problem.

strengths:
1. The paper is well-written and clear.

2. Leveraging deep generative models to learn the uncertainty set is a promising approach.

weaknesses:
1. The Projected Gradient Ascent (PGA) method does not guarantee convergence to the worst-case uncertainty realization. Although the authors propose randomly initializing PGA with different samples of $z$ for empirical performance, providing some theoretical analysis on the approximation error would be beneficial.

2. The performance improvement of the proposed method is minimal.

questions:
1. The paper suggests that the framework is general and could also be applied to diffusion models. However, diffusion model training involves matching the score of the noised distribution, and samples cannot be easily obtained during training. Could you elaborate on how diffusion models would integrate with your proposed framework?

2. How do you ensure the uncertainty set learned by the VAE is sufficiently tight? When optimizing over the latent space, are there any constraints? If not, is there a risk that the algorithm could select an overly conservative worst-case realization?
</content>
</thread_object>

</thread>
<thread>
creation date:
1731020948163

thread_type:
official_review

<thread_object>
creator: reviewer_foz7

<content>
summary:
This paper presents a novel deep generative approach using Variational Autoencoders (VAE) to tackle two-stage adaptive robust optimization (ARO) under high-dimensional uncertainty. Traditional ARO approaches for constructing the uncertainty set $\mathcal{U}$ tend to be overly conservative, often leading to excessive resource allocation in scenarios with high-dimensional and irregularly distributed uncertainties. The proposed AGRO method mitigates this issue by incorporating VAE embedding and column-and-constraint generation (CCG). The method also uses projected gradient ascent to solve the formulated subproblem. Experiments on two problems demonstrate the advantages of this method over conventional CCG approaches.

strengths:
The primary contribution of this work is the innovative application of VAE to construct a tighter uncertainty set, thereby reducing over-conservatism in high-dimensional decision-making, which is then addressed through CCG. The paper is clearly presented, with informative visuals such as Figure 2, and well-organized notation and formulations. The experimental results highlight the promise of the proposed method.

weaknesses:
The proposed approach involves training a VAE, whose performance might be sensitive to hyperparameters, computational resources, and the amount of available training data. Additional experiments and discussion could enhance the paper’s applicability. Please see the following questions for further details.

questions:
- In Figure 1, the authors use two 3D visualizations to illustrate the two-stage operations on $\mathcal{U}$ and $\mathcal{Z}$. Could the authors provide a brief description of the specific main problem addressed here to give the audience a better understanding?

- In the experiment on the production-distribution problem, the authors observed a reverse effect of the bottleneck dimension in the low-dimensional case of $|\mathcal{J}|=3$. Could the authors elaborate on possible reasons for this?

- Based on the experiments, could the authors provide practical guidelines for selecting the appropriate bottleneck dimension for VAEs according to the dimensionality or complexity of the uncertainty set?

- For tabular results such as those in Figure 3 (left) and Table 1, could the authors also report the standard deviation across trials? This would help readers understand the robustness of AGRO in different practical scenarios.

- Could the authors provide more details on the VAE architecture and training settings used in each experiment? Such as layer dimensions, normalization, optimizer, and learning rate,  for better reproducibility.
</content>
</thread_object>

</thread>
<thread>
creation date:
1731405272946

thread_type:
official_review

<thread_object>
creator: reviewer_vrq2

<content>
summary:
This paper introduces AGRO, a novel method for two-stage adaptive robust optimization (ARO) using a variational autoencoder (VAE) to generate adversarial and realistic uncertainty sets. The authors demonstrate that AGRO reduces planning costs in ARO tasks, outperforming classical approaches.

strengths:
1. The proposed AGRO framework is innovative, embedding a VAE within a column-and-constraint generation (CCG) scheme to achieve high-dimensional adversarial generation with cost efficiency.
2. It seems that the empirical results highlight an over 10% cost reductions over classical methods.

weaknesses:
1.	The introduction lacks a comprehensive motivation for using a VAE for uncertainty sets over other generative models. The authors should justify why a VAE was chosen and discuss the potential advantages over alternatives, like GANs or normalizing flows, which may also be suitable.
2.	The discussion on the choice of the VAE bottleneck dimension (parameter L) could be expanded. The authors should provide more insight into how different L values affect the uncertainty set’s coverage and the balance between computational cost and model fidelity.
3.	While the experiments are detailed, there is no mention of computational time for VAE training or comparison with other ARO solutions. Including such results would enhance transparency about AGRO’s feasibility in larger-scale applications.
4.	The paper does not explore alternative formulations for the adversarial subproblem. A comparison with different optimization methods or a discussion on the limitations of projected gradient ascent could further clarify AGRO's robustness.

questions:
1.	Why did you choose a VAE over other generative models (e.g., GANs, normalizing flows) for constructing uncertainty sets in AGRO? Would these models offer any advantages or limitations compared to VAEs in this application?
2.	How does the bottleneck dimension (L) influence the overall performance and reliability of AGRO? Could you elaborate on any trade-offs between computational cost and uncertainty set coverage as L varies?
3.	The paper discusses using VAE-based uncertainty sets to achieve tighter approximations. Could you clarify how you ensure these sets are both realistic and adversarial? Are there any specific quantitative or qualitative metrics that assess the accuracy of these generated uncertainty sets?
</content>
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</thread>
<thread>
creation date:
1733161857257

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Final note to reviewers

comment:
As we approach the conclusion of the discussion period, we invite reviewers to share any final questions or comments for the authors before proceeding with their evaluations. Below is an overview of new findings and major revisions made to the final submitted manuscript:

### Revised Deterministic Parameters in Section 4
We have updated the deterministic parameters used to define the capacity expansion problem in Section 4, enhancing the realism of the capacity expansion model. This adjustment is reflected in the latest experimental findings, where the case of $L=4$ now yields the lowest cost across all methods instead of $L=2$. This shift highlights the influence of ramping and net-load volatility on operational and overall planning costs. The revised parameters also resulted in a less tractable bilinear subproblem for CCG, leading to a sixfold increase in runtime. These refinements provide a more faithful representation of capacity expansion planning considerations, ultimately strengthening our experimental results.

### New Subsection in Section 4.2
We have added a subsection titled "Bottleneck Dimension" to Section 4.2. This subsection connects insights on the role of the bottleneck dimension $L$ from the production-distribution case study to those from the capacity expansion planning case study. Additionally, it includes a paragraph offering recommendations for selecting $L$.

### Updates to Section 3.1
The second paragraph of Section 3.1 has been revised to highlight the potential advantages of GANs, normalizing flows, and diffusion models in generating higher-fidelity samples compared to VAEs. We then justify the use of VAEs by referencing experimental findings in Section 4, which show that higher generative fidelity does not necessarily lead to better performance concerning the ARO objective.

### Expanded Computational Experiment Details
Additional details have been included regarding the computational experiments. Specifically, Figure 3 (left) and Table 1 now report standard deviations across all trials. Furthermore, Appendices B.1 and B.2 have been expanded to provide more comprehensive information about the architecture and training process for the VAEs.

### Comparison of PGA with Direct Solve
Finally, we note that the final submission does not include a complete comparison of projected gradient ascent (PGA) with the direct solve approach for the mixed-binary bilinear formulation of the AGRO subproblem (see Appendix A.2). However, our experimental findings can be summarized as follows:

- For the production-distribution problem with $|\mathcal{J}| \in \{3,6\}$, PGA obtained solutions to the adversarial subproblem that were, on average, within 1\% of optimality as determined by the direct solve.

- For the capacity expansion planning problem, a full cost comparison could not be provided as the direct solve failed to achieve provably optimal solutions within a reasonable timeframe. Specifically, the solver reported a 400\% optimality gap after one hour.

***

### Thank you!
We are grateful for your thoughtful feedback throughout this process. We look forward to your final evaluations and appreciate your valuable contributions to improving this manuscript.
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Metareview:
metareview:
This paper introduces AGRO, a two-stage adaptive robust optimization (ARO) method that combines variational autoencoder (VAE) within a column-and-constraint generation for adversarial uncertainty sets. The authors show that AGRO reduces planning costs compared to classical ARO approaches. 

This is indeed a borderline paper. Although the proposed method achieves significant cost reduction, the exploiting generative model in learning for decision making and RL is not novel, and the choice of VAE seems not well justified.