  We consider convex constrained optimization problems that also include a
cardinality constraint. In general, optimization problems with cardinality
constraints are difficult mathematical programs which are usually solved by
global techniques from discrete optimization. We assume that the region defined
by the convex constraints can be written as the intersection of a finite
collection of convex sets, such that it is easy and inexpensive to project onto
each one of them (e.g., boxes, hyper-planes, or half-spaces).
  Taking advantage of a recently developed continuous reformulation that
relaxes the cardinality constraint, we propose a specialized penalty gradient
projection scheme combined with alternating projection ideas to solve these
problems. To illustrate the combined scheme, we focus on the standard
mean-variance portfolio optimization problem for which we can only invest in a
preestablished limited number of assets.
  For these portfolio problems with cardinality constraints we present a
numerical study on a variety of data sets involving real-world capital market
indices from major stock markets. On those data sets we illustrate the
computational performance of the proposed scheme to produce the effective
frontiers for different values of the limited number of allowed assets.

Optimization and Control