Optimal portfolio selections via $$\ell _{1, 2}$$ ℓ 1 , 2 -norm regularization

There has been much research about regularizing optimal portfolio selections through $$\ell _1$$ ℓ 1 norm and/or $$\ell _2$$ ℓ 2 -norm squared. The common consensuses are (i) $$\ell _1$$ ℓ 1 leads to sparse portfolios and there exists a theoretical bound that limits extreme shorting of assets; (ii) $$\ell _2$$ ℓ 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 $$\ell _1$$ ℓ 1 and $$\ell _2$$ ℓ 2 norm combined (namely $$\ell _{1, 2}$$ ℓ 1 , 2 -norm). The new regularization enjoys the best of the two regularizations of $$\ell _1$$ ℓ 1 norm and $$\ell _2$$ ℓ 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 $$\ell _{1,2}$$ ℓ 1 , 2 norm. A fast proximal augmented Lagrange method is applied to solve the $$\ell _{1,2}$$ ℓ 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.