  In this paper we propose and discuss different 0-1 linear models in order to
solve the cardinality constrained portfolio problem by using factor models.
Factor models are used to build portfolios to track indexes, together with
other objectives, also need a smaller number of parameters to estimate than the
classical Markowitz model. The addition of the cardinality constraints limits
the number of securities in the portfolio. Restricting the number of securities
in the portfolio allows us to obtain a concentrated portfolio, reduce the risk
and limit transaction costs. To solve this problem, a pure 0-1 model is
presented in this work, the 0-1 model is constructed by means of a piecewise
linear approximation. We also present a new quadratic combinatorial problem,
called a minimum edge-weighted clique problem, to obtain an equality weighted
cardinality constrained portfolio. A piecewise linear approximation for this
problem is presented in the context of a multi factor model. For a single
factor model, we present a fast heuristic, based on some theoretical results to
obtain an equality weighted cardinality constraint portfolio. The consideration
of a piecewise linear approximation allow us to reduce significantly the
computation time required for the equivalent quadratic problem. Computational
results from the 0-1 models are compared to those using a state-of-the-art
Quadratic MIP solver.
