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Case study background and problem formulations
The optimization problems are solved using standard Python libraries (e.g., SciPy) and advanced convex/mixed-integer solvers (e.g., GUROBI). Please refer to the Environment and Python Framework Instruction.
PROBLEM: Mixture Quantile Estimation for Electricity Price Data
Minimize L1 / L2 error with Non-negativity, L1 Selection Penalty, and Smoothness Constraints.
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This case study demonstrates the formulation and solution of the Mixture Quantile (MQ) model applied to daily U.S. electricity prices. Unlike traditional Maximum Likelihood Estimation, this framework reduces distribution fitting to a constrained linear regression problem. The Python implementation utilizes parametric bases (Standard Gaussian, Student’s t), Tail-specific Generalized Pareto Distributions (GPD) for extreme price spikes, and non-parametric I-splines for flexible body fitting.
Data and solution in Python Environment
| Problem Dataset | # of Variables | # of Samples | ||
|---|---|---|---|---|
| U.S. Electricity Prices (2001-2024) | Python Code | Solution | 19 (Basis functions) | 79,651 |
References
• Peng, C., Li, Y. and S. Uryasev. Mixture Quantiles Estimated by Constrained Linear Regression. Annals of Operations Research. 2026, Accepted for publication. https://uryasev.ams.stonybrook.edu/wp-content/uploads/2026/04/Mixture_quantiles_constrained_linear_regression.pdf.