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# Optimal portfolio selections via ℓ 1 , 2 -norm regularization

Zhao, Hongxin, Kong, LingChen and Qi, Hou-Duo (2021) Optimal portfolio selections via ℓ 1 , 2 -norm regularization. Computational Optimization and Applications, 80 (3), 853-881.

Record type: Article

## Abstract

There has been much research about regularizing optimal portfolio selections through $\ell_1$ norm and/or $\ell_2$-norm squared. The common consensuses are (i) $\ell_1$ leads to sparse portfolios and there exists a theoretical bound that limits extreme shorting of assets; (ii) $\ell_2$ (norm-squared) stabilizes the computation by improving the condition number of the problem resulting in strong out-of-sample performance; and (iii) there exist efficient numerical algorithms for those regularized portfolios with closed-form solutions each step.
When combined such as in the well-known elastic net regularization, theoretical bounds are difficult to derive so as to limit extreme shorting of assets. In this paper, we propose a minimum variance portfolio with the regularization of $\ell_1$ and $\ell_2$ norm combined (namely $\ell_{1, 2}$-norm). The new regularization enjoys the best of the two regularizations of $\ell_1$ norm and $\ell_2$-norm squared. In particular, we derive a theoretical bound that limits short-sells and develop a closed-form formula for the proximal term of the $\ell_{1,2}$ norm. A fast proximal augmented Lagrange method is applied to solve the $\ell_{1,2}$-norm regularized problem. Extensive numerical experiments confirm that the new model often results in high Sharpe ratio, low turnover and small amount of short sells when compared with several existing models on six datasets.

Text
mvpl12_for_PURE - Accepted Manuscript

Accepted/In Press date: 20 August 2021
e-pub ahead of print date: 15 September 2021
Published date: December 2021
Additional Information: Funding Information: The paper was supported in part by 111 Project of China (B16002), IEC/NSFC/191543 and NSFC(12071022)
Keywords: Minimum variance portfolio, Out-of-sample performance, Portfolio optimization, Proximal augmented Lagrange method, ℓ -norm regularization

## Identifiers

Local EPrints ID: 451606
URI: http://eprints.soton.ac.uk/id/eprint/451606
ISSN: 0926-6003
PURE UUID: 65db5809-e972-49af-9afe-68c6314534f0
ORCID for Hou-Duo Qi: orcid.org/0000-0003-3481-4814

## Catalogue record

Date deposited: 14 Oct 2021 16:30

## Contributors

Author: Hongxin Zhao
Author: LingChen Kong
Author: Hou-Duo Qi