# Constrained Max Drawdown: a Fast and Robust Portfolio Optimization Approach

**Authors:** Albert Dorador  
**arXiv:** [2401.02601](https://arxiv.org/abs/2401.02601)  
**Date:** 2024-01-05  
**Summary page:** https://www.summarizepaper.com/en/paper/2401.02601v1/

## Key points

- Analysis of portfolio optimization models
- Proposal of a faster and more robust model with comparable return-to-risk ratio
- Examination of classical Markowitz model and two variations, including additional analysis on quadratic programming (QP) models
- Confirmation of Strong Duality through verification of Slater’s conditions
- Demonstration that L1 regularization has no impact on portfolio optimization without short-selling
- Introduction of a heuristic for selecting the λ parameter when optimizing mean and variance simultaneously
- Introduction of a fourth model from Konno and Yamazaki (1991) utilizing linear programming (LP) formulation in portfolio optimization
- Discussion on the 'epigraph trick' used to transform non-linear objectives into LP in the new model
- Presentation of a new model offering an alternative linearization of classical QP formulation with significant speed-up and performance improvements
- Brief mention of incorporating downside risk measures in the model
- Proposal for a variation transforming LP into mixed-integer linear programming (MILP) with minimum allocations constraint, resulting in faster solutions and greater robustness compared to other models
- In-depth analysis conducted on optimal solution provided by MILP formulation, highlighting top three stock choices during challenging times like COVID-19.

## Summary

This paper analyzes various portfolio optimization models and proposes a faster and more robust model that maintains a comparable ratio of portfolio expected return to risk. The study delves into the classical Markowitz model and two well-known variations, providing additional analysis on these quadratic programming (QP) models. The research confirms Strong Duality by verifying Slater’s conditions and demonstrates that L1 regularization has no impact on portfolio optimization without short-selling. Additionally, a heuristic for objectively selecting the λ parameter when optimizing mean and variance simultaneously is introduced. Furthermore, the paper introduces a fourth model from Konno and Yamazaki (1991), one of the earliest examples of linear programming (LP) formulation in portfolio optimization. The model's utilization of the 'epigraph trick' to transform a non-linear objective into an LP is discussed along with its theoretical advantages over traditional QP formulations. The paper then presents a new model that offers an alternative linearization of the classical QP formulation. Initially presented as a maximin problem before being converted into an LP using the epigraph trick, this model showcases nearly tenfold speed-up compared to traditional QPs while maintaining similar or improved in-sample and out-of-sample performance. The benefits of incorporating downside risk measures in the model are briefly touched upon. A variation of this new model is proposed, transforming the LP into a mixed-integer linear programming (MILP) by imposing minimum allocations of 5% for any budget allocated to a specific stock. Despite being theoretically more challenging to solve due to NP-completeness, practical implementation results in a 25 times faster solution than the LP formulation and 200 times faster than QPs while delivering competitive performance metrics. Notably, the MILP formulation exhibits greater robustness compared to other models when faced with parameter perturbations. With the MILP formulation emerging as the most compelling option, an in-depth analysis of its optimal solution is conducted, highlighting why its top three stock choices are deemed reasonable during times like COVID-19. This comprehensive exploration aims to contribute valuable insights for future research in portfolio optimization methodologies.