(125g) Distributionally Robust Optimization for Gaussian Mixture Model Ambiguity Under Moment Variations

In conventional deterministic optimization problems, it is typically assumed that all model parameters are precisely known and fixed. However, this assumption is often unrealistic in practice, as many parameters may be unknown or difficult to predict accurately, and they can fluctuate over time or be influenced by random factors.[1] Assuming uncertain parameters are known with certainty often leads to infeasible or suboptimal decisions when implemented in real-world situations.

Distributionally robust optimization (DRO) methods have gained significant attention as effective approaches for addressing uncertainty in solving optimization problems. It aims to determine an optimal solution that remains effective under the worst-case distribution within the ambiguity set. The Wasserstein DRO method[2], which does not impose assumptions on the properties of candidate distributions within the ambiguity set, has been extensively studied. However, in some cases, a Gaussian mixture model (GMM) serves as an effective representation of the distribution of uncertain parameters. The existing W_d ^weight DRO method[3] uses this property by constructing an ambiguity set around a GMM, where the Gaussian components of the candidate distributions are fixed, and variability arises from the component weights. But this approach may suffer from inaccuracies in the specified Gaussian components and risks insufficient robustness when ambiguity set saturation occurs.

In this work, we propose a complementary GMM-based DRO approach W_d ^moment that specifies the component weights of the candidate GMM distributions while allowing variability in the Gaussian components. The tractable formulation is developed using the McCormick envelope, and an iterative bound tightening algorithm is applied. The proposed method is compared with Wasserstein DRO and W_d ^weight DRO. The results show that the proposed DRO method achieves a favorable balance between conservatism and robustness.

[1] Balasubramanian, J.; Grossmann, I. E. Scheduling optimization under uncertainty—an alternative approach. Computers & Chemical Engineering 2003, 27, 469–490.

[2] Kuhn, Daniel, et al. "Wasserstein distributionally robust optimization: Theory and applications in machine learning." Operations research & management science in the age of analytics. Informs, 2019. 130-166.

[3] Kammammettu, S.; Yang, S.-B.; Li, Z. Distributionally robust optimization using optimal transport for Gaussian mixture models. Optimization and Engineering 2023, 1–26.