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

TL;DR

This paper introduces a scalable gradient-based optimization framework for sparse minimum-variance portfolio selection, transforming a combinatorial problem into a continuous one, enabling faster solutions with comparable accuracy to traditional solvers.

Contribution

The authors develop a novel Boolean relaxation approach that efficiently solves sparse portfolio optimization problems, improving scalability over existing mixed-integer quadratic programming methods.

Findings

Matches commercial solvers in asset selection accuracy

Operates efficiently on larger problem sizes

Achieves near-optimal portfolio variance with fewer assets

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.