Code for the sparse index tracking problem. We take a user constructed index of stock symbols, and train a model on the return matrix of the stocks in the index. over a given date range. The goal is to compute a portfolio consisting of a sparse subset of the stocks in the index that will track the index with small tracking error. That is, we're trying to find a quick approximation of the problem:

min_w ||Rw - y||_2 + \lambda*card(w) subject to w \geq 0, \sum_i w_i = 1

where R is an mxn matrix of returns. n is the number of assets in the index. m the time periods of the training set So the ith column is the vector of returns for the ith asset.

w is a length n vector of weights. w_i represents the proportion of the portfolio invested in asset i.

y is a length m vector representing the returns for the composit index over the training period

card(w) maps w to its cardinality.

lambda is a regularization parameter that penalizes the cardinality of the solution.

The constraint w \geq 0 means we do not allow short selling \sum_i w_i =1 is a normalization constraint (We have $1 to invest, and we're investing it all).