Selection of the securities for investment portfolio
design is one of the most important optimization
problems of the last century. For this, numerous
strategies and mathematical models have been
proposed. For instance, the passive investment
strategy performs the tracking of market indices with the
intention of reproducing its performance with an
optimized portfolio as described in [1].

This passive strategy is based on the advances shown
by Palomar [2] who deals with the issue of designing
sparse portfolios to efficiently reproduce the returns of
any index. Once the stocks have been selected, the
following step aims at dividing the investment capital
between these stocks in some efficient way. This
strategy has shown promising performance, however, it
does not take into account the correlation between the
selected stock returns, which is an important factor in
the efficient selection of the stocks, but a cointegration
based approach.

Therefore, the main objective of this work relies on
formulating a mathematical model that allows to find
high correlated stocks for the sparse portfolio design.
Thus, it aims at modifying previous work to improve the
quality results by taking into account the correlation
between the stocks.

In this manner, the proposed optimization problem
includes the nuclear norm over the market returns
matrix multiplied by the desired variable weights, such
that it is possible to apply some thresholding technique
over the singular value decomposition of this resulting
matrix as presented in [3]. This allows to reduce its rank
iteratively with the objective of obtaining its low-rank
approximation, which multiplied by the inverse returns
matrix, results in the desired portfolio weights