Belief Propagation Algorithm for Portfolio Optimization Problems

TL;DR

This paper introduces a belief propagation algorithm for portfolio optimization, validating its effectiveness through replica analysis and confirming the similarity of solutions across different models.

Contribution

It develops an approximate belief propagation method for portfolio optimization and verifies its consistency with replica analysis, advancing solution techniques in this domain.

Findings

The belief propagation algorithm aligns with replica analysis results.

Optimal solutions for absolute deviation and mean-variance models exhibit similar behavior.

The proposed method provides an efficient approximation for complex portfolio problems.

Abstract

The typical behavior of optimal solutions to portfolio optimization problems with absolute deviation and expected shortfall models using replica analysis was pioneeringly estimated by S. Ciliberti and M. M\'ezard [Eur. Phys. B. 57, 175 (2007)]; however, they have not yet developed an approximate derivation method for finding the optimal portfolio with respect to a given return set. In this study, an approximation algorithm based on belief propagation for the portfolio optimization problem is presented using the Bethe free energy formalism, and the consistency of the numerical experimental results of the proposed algorithm with those of replica analysis is confirmed. Furthermore, the conjecture of H. Konno and H. Yamazaki, that the optimal solutions with the absolute deviation model and with the mean-variance model have the same typical behavior, is verified using replica analysis and the belief propagation algorithm.