A Scalable Gradient-Based Optimization Framework for Sparse Minimum-Variance Portfolio Selection

ArXiv ID: 2505.10099 “View on arXiv”

Authors: Sarat Moka, Matias Quiroz, Vali Asimit, Samuel Muller

Abstract

Portfolio optimization involves selecting asset weights to minimize a risk-reward objective, such as the portfolio variance in the classical minimum-variance framework. Sparse portfolio selection extends this by imposing a cardinality constraint: only $k$ assets from a universe of $p$ may be included. The standard approach models this problem as a mixed-integer quadratic program and relies on commercial solvers to find the optimal solution. However, the computational costs of such methods increase exponentially with $k$ and $p$, making them too slow for problems of even moderate size. We propose a fast and scalable gradient-based approach that transforms the combinatorial sparse selection problem into a constrained continuous optimization task via Boolean relaxation, while preserving equivalence with the original problem on the set of binary points. Our algorithm employs a tunable parameter that transmutes the auxiliary objective from a convex to a concave function. This allows a stable convex starting point, followed by a controlled path toward a sparse binary solution as the tuning parameter increases and the objective moves toward concavity. In practice, our method matches commercial solvers in asset selection for most instances and, in rare instances, the solution differs by a few assets whilst showing a negligible error in portfolio variance.

Keywords: portfolio optimization, sparse portfolio selection, mixed-integer quadratic program, gradient-based approach, Boolean relaxation, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper employs advanced mathematical concepts like Boolean relaxation, convex-concave duality, and Frank-Wolfe algorithms, indicating high math complexity, but presents only theoretical guarantees and synthetic/real-world dataset experiments without code or detailed backtesting, placing it in the Lab Rats quadrant.

  flowchart TD
    A["Research Goal: Fast Sparse Minimum-Variance Portfolio Selection"] --> B["Methodology: Boolean Relaxation & Gradient Optimization"]
    B --> C["Input: p Assets, Data, Constraint k"]
    C --> D{"Algorithm"}
    D --> E["Phase 1: Tune Parameter λ<br/>Convex Objective"]
    E --> F["Phase 2: Increase λ<br/>Path to Concavity"]
    F --> G["Sparse Binary Solution<br/>(Portfolio Weights)"]
    G --> H["Outcome: Matches Commercial Solvers<br/>Scalable & Fast"]