  A cardinality-constrained portfolio caps the number of stocks to be traded
across and within groups or sectors. These limitations arise from real-world
scenarios faced by fund managers, who are constrained by transaction costs and
client preferences as they seek to maximize return and limit risk.
  We develop a new approach to solve cardinality-constrained portfolio
optimization problems, extending both Markowitz and conditional value at risk
(CVaR) optimization models with cardinality constraints. We derive a continuous
relaxation method for the NP-hard objective, which allows for very efficient
algorithms with standard convergence guarantees for nonconvex problems. For
smaller cases, where brute force search is feasible to compute the globally
optimal cardinality- constrained portfolio, the new approach finds the best
portfolio for the cardinality-constrained Markowitz model and a very good local
minimum for the cardinality-constrained CVaR model. For higher dimensions,
where brute-force search is prohibitively expensive, we find feasible
portfolios that are nearly as efficient as their non-cardinality constrained
counterparts.
