Solving norm constrained portfolio optimization via coordinate-wise descent algorithms

A fast method based on coordinate-wise descent algorithms is developed to solve portfolio optimization problems in which asset weights are constrained by lq norms for 1≤q≤2. The method is first applied to solve a minimum variance portfolio (mvp) optimization problem in which asset weights are constrained by a weighted l1 norm and a squared l2 norm. Performances of the weighted norm penalized mvp are examined with two benchmark data sets. When the sample size is not large in comparison with the number of assets, the weighted norm penalized mvp tends to have a lower out-of-sample portfolio variance, lower turnover rate, fewer numbers of active constituents and shortsale positions, but higher Sharpe ratio than the one without such penalty. Several extensions of the proposed method are illustrated; in particular, an efficient algorithm for solving a portfolio optimization problem in which assets are allowed to be chosen grouply is derived.