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

TL;DR

This work tackles sparse minimum-variance portfolio selection, a combinatorial optimization problem that is NP-hard when enforcing a cardinality constraint. It develops a gradient-based approach by reframing the problem through Boolean relaxation, introducing a tunable parameter $\delta$ that shifts the auxiliary objective from convex to concave, and solving via a Grid-FW continuation of Frank-Wolfe updates. The authors prove convergence properties and solution-continuity as $\delta$ increases, and demonstrate scalability with a gradient-based algorithm whose complexity scales as $O(n m p^2)$, enabling efficient handling of large asset universes. Empirical results on synthetic and real-world data show that Grid-FW achieves near-optimal or optimal variance on moderate cases and substantially outperforms exact solvers on large-scale problems, offering a practical, scalable alternative for sparse portfolio construction.

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.

Figures (1)

• Figure 1: Panel (a): Surface plots of the auxiliary objective function $f_\delta(\boldsymbol{t})$ for a $2\times 2$-dimensional covariance matrix $\Sigma$ for different values of $\delta$, displaying convexity for $\delta = 0.005$ and concavity for $\delta = 5$. The values of the function at the corners $\boldsymbol{s} \in \{0, 1\}^2$ correspond to those of the discrete function $-\boldsymbol{1}^\top \Sigma_{[\boldsymbol{s}]}^{-1}\boldsymbol{1}$ in \ref{['eqn:opt-bc3']}. Panel (b): Iterative convergence of $t_j$'s toward $0$ or $1$ for a dataset of $p=31$ assets when $k = 4$ during the execution of Algorithm \ref{['alg:alg']}. The paths of the four $t_j$ that converge to $1$ are shown in black, while other paths are shown in grey.

Theorems & Definitions (17)

• Lemma 1
• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Theorem 4
• Remark 1
• Remark 2
• proof : Proof of Lemma \ref{['lem:key-res2']}
• proof : Proof of Theorem \ref{['thm:monotonicity']}