Enhancing Numerical Stability in Portfolio Optimization via Numerical Rank: The $$\delta $$-Stable Part of the Efficient Frontier

Abstract

Abstract The fact that the solution of the Markowitz model is numerically unstable is recognized as one of the principal reasons for its poor out-of-sample results when applied in practice. In the present paper, we show that, alongside standard instability sources such as the estimation of the covariance matrix and returns, there is a structural source of instability. This source depends on the geometry of the admissible region defined by the linear constraints of the optimization problem namely, the budget constraint and the target expected return. We provide theoretical results based on numerical arguments and propose a set of suitable target returns that restricts the standard mean-variance efficient frontier to a subset of portfolios that are numerically stable with respect to a given parameter $$\delta $$ δ . The increased stability of these portfolios is achieved by imposing a condition that preserves the numerical rank of the constraint matrix. An application to various real-world financial databases highlights the effectiveness of our proposal in reducing the numerical instability of the optimal solution.

The authors' abstract, as published at the source. Operations Research Forum, 2026 · DOI ↗