Bilevel robust optimization approach for multi-period sparse portfolio selection

Abstract

Portfolio optimization focuses on efficiently allocating investment across assets, a process often impacted by input parameter uncertainties. Robust optimization enhances traditional portfolio optimization models by accounting for these uncertainties, aiming to find solutions that perform well under a wide range of possible scenarios. We propose a robust version of the multi-period sparse mean–variance model where the uncertainty on the covariance matrix is described using a box uncertainty set. The bi-level worst-case problem is reformulated as a convex non-smooth single-level one by replacing the lower level with its Karush–Kuhn–Tucker conditions. An effective method for solving this problem is the alternating direction method of multipliers, which is characterized by theoretical solid convergence properties, even when the resulting subproblems are solved approximately. Numerical tests on real market data illustrate a good trade-off between optimality and robustness.