The Boosted Difference of Convex Functions Algorithm for Value-at-Risk Constrained Portfolio Optimization

Paper: arXiv 2402.09194

Abstract

A highly relevant problem of modern finance is the design of Value-at-Risk (VaR) optimal portfolios. Due to contemporary financial regulations, banks and other financial institutions are tied to use the risk measure to control their credit, market, and operational risks. Despite its practical relevance, the non-convexity induced by VaR constraints in portfolio optimization problems remains a major challenge. To address this complexity more effectively, this paper proposes the use of the Boosted Difference-of-Convex Functions Algorithm (BDCA) to approximately solve a Markowitz-style portfolio selection problem with a VaR constraint. As one of the key contributions, we derive a novel line search framework that allows the application of the algorithm to Difference-of-Convex functions (DC) programs where both components are non-smooth. Moreover, we prove that the BDCA linearly converges to a Karush-Kuhn-Tucker point for the problem at hand using the Kurdyka-Lojasiewicz property. We also outline that this result can be generalized to a broader class of piecewise-linear DC programs with linear equality and inequality constraints. In the practical part, extensive numerical experiments under consideration of best practices then demonstrate the robustness of the BDCA under challenging constraint settings and adverse initialization. In particular, the algorithm consistently identifies the highest number of feasible solutions even under the most challenging conditions, while other approaches from chance-constrained programming lead to a complete failure in these settings. Due to the open availability of all data sets and code, this paper further provides a practical guide for transparent and easily reproducible comparisons of VaR-constrained portfolio selection problems in Python.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.0/10 (capped from 7.5: transaction costs not modelled; no out-of-sample test; see penalty flags)
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced mathematical theory including Difference of Convex functions, Kurdyka-Łojasiewicz property, and non-smooth analysis, while also presenting extensive empirical experiments with open-source code and data, demonstrating practical implementation readiness.

Research Flowchart

  flowchart TD
  A["Research Goal: Solve VaR-constrained<br>portfolio optimization"] --> B["Methodology: Apply BDCA with<br>novel line search for non-smooth DC"]
  B --> C["Inputs: Historical asset data<br>via open-source datasets"]
  C --> D["Process: Run BDCA & competitors<br>under challenging constraints"]
  D --> E{"Analysis: Verify convergence<br>via KKT/KL conditions"}
  E --> F["Outcomes: BDCA yields robust,<br>feasible portfolios consistently"]

Cited References (84)

1. A portfolio optimization model with three objectives and discrete variables (2010)
2. Accelerating the DC algorithm for smooth functions (2018)
3. The Boosted Difference of Convex Functions Algorithm For Nonsmooth Functions (2020)
4. The Boosted DC Algorithm for Linearly Constrained DC Programming (2022)
5. When to use the Bonferroni correction (2014)
6. Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka- \L ojasiewicz Inequality (2010)
7. A novel multi period mean-VaR portfolio optimization model considering practical constraints and transaction cost (2019)
8. Bootstrap Hypothesis Testing Procedures (1993)

0 of 84 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.