  This paper analyzes the statistical properties of constrained portfolio
formation in a high dimensional portfolio with a large number of assets.
Namely, we consider portfolios with tracking error constraints, portfolios with
tracking error jointly with weight (equality or inequality) restrictions, and
portfolios with only weight restrictions. Tracking error is the portfolio's
performance measured against a benchmark (an index usually), {\color{black}{and
weight constraints refers to specific allocation of assets within the
portfolio, which often come in the form of regulatory requirement or fund
prospectus.}} We show how these portfolios can be estimated consistently in
large dimensions, even when the number of assets is larger than the time span
of the portfolio. We also provide rate of convergence results for weights of
the constrained portfolio, risk of the constrained portfolio and the Sharpe
Ratio of the constrained portfolio. To achieve those results we use a new
machine learning technique that merges factor models with nodewise regression
in statistics. Simulation results and empirics show very good performance of
our method.
