A Relaxed Optimization Approach for Cardinality-Constrained Portfolio Optimization

TL;DR

This paper introduces a continuous relaxation method for efficiently solving NP-hard cardinality-constrained portfolio optimization problems, extending Markowitz and CVaR models with guarantees and near-optimal solutions.

Contribution

It develops a novel relaxation approach that enables efficient algorithms for cardinality-constrained portfolio optimization, applicable to both small and large problem sizes.

Findings

Efficient algorithms with convergence guarantees for nonconvex problems.

Global optimality for small cases using brute-force search.

Near-optimal feasible portfolios for high-dimensional problems.

Abstract

A cardinality-constrained portfolio caps the number of stocks to be traded across and within groups or sectors. These limitations arise from real-world scenarios faced by fund managers, who are constrained by transaction costs and client preferences as they seek to maximize return and limit risk. We develop a new approach to solve cardinality-constrained portfolio optimization problems, extending both Markowitz and conditional value at risk (CVaR) optimization models with cardinality constraints. We derive a continuous relaxation method for the NP-hard objective, which allows for very efficient algorithms with standard convergence guarantees for nonconvex problems. For smaller cases, where brute force search is feasible to compute the globally optimal cardinality- constrained portfolio, the new approach finds the best portfolio for the cardinality-constrained Markowitz model and a very good local minimum for the cardinality-constrained CVaR model. For higher dimensions, where brute-force search is prohibitively expensive, we find feasible portfolios that are nearly as efficient as their non-cardinality constrained counterparts.