  We consider general nonlinear programming problems with cardinality
constraints. By relaxing the binary variables which appear in the natural
mixed-integer programming formulation, we obtain an almost equivalent nonlinear
programming problem, which is thus still difficult to solve. Therefore, we
apply a Scholtes-type regularization method to obtain a sequence of easier to
solve problems and investigate the convergence of the obtained KKT points. We
show that such a sequence converges to an S-stationary point, which corresponds
to a local minimizer of the original problem under the assumption of convexity.
  Additionally, we consider portfolio optimization problems where we minimize a
risk measure under a cardinality constraint on the portfolio. Various risk
measures are considered, in particular Value-at-Risk and Conditional
Value-at-Risk under normal distribution of returns and their robust
counterparts under moment conditions. For these investment problems formulated
as nonlinear programming problems with cardinality constraints we perform a
numerical study on a large number of simulated instances taken from the
literature and illuminate the computational performance of the Scholtes-type
regularization method in comparison to other considered solution approaches: a
mixed-integer solver, a direct continuous reformulation solver and the
Kanzow-Schwartz regularization method, which has already been applied to
Markowitz portfolio problems.
