Random matrix approach for primal-dual portfolio optimization problems

TL;DR

This paper applies random matrix theory and Lagrange multipliers to analyze primal and dual portfolio optimization problems with identical asset variances, providing new insights and validation through numerical experiments.

Contribution

It introduces a novel application of the random matrix approach to primal-dual portfolio optimization problems with identical variances, comparing results with previous methods.

Findings

Validated the random matrix approach against numerical experiments.

Provided analytical solutions for primal and dual problems.

Compared new results with previous work to highlight differences.

Abstract

In this paper, we revisit the portfolio optimization problems of the minimization/maximization of investment risk under constraints of budget and investment concentration (primal problem) and the maximization/minimization of investment concentration under constraints of budget and investment risk (dual problem) for the case that the variances of the return rates of the assets are identical. We analyze both optimization problems by using the Lagrange multiplier method and the random matrix approach. Thereafter, we compare the results obtained from our proposed approach with the results obtained in previous work. Moreover, we use numerical experiments to validate the results obtained from the replica approach and the random matrix approach as methods for analyzing both the primal and dual portfolio optimization problems.