A Scalable Gradient-Based Optimization Framework for Sparse Minimum-Variance Portfolio Selection
This addresses the scalability issue in portfolio optimization for financial practitioners, offering a more efficient method for moderate-sized problems, though it is incremental as it builds on existing relaxation techniques.
The paper tackles the computationally expensive problem of sparse minimum-variance portfolio selection by proposing a fast gradient-based optimization framework that transforms it into a constrained continuous task via Boolean relaxation, achieving solutions that match commercial solvers in most instances with negligible variance error.
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.