  In this paper, we aim at solving the cardinality constrained high-order
portfolio optimization, i.e., mean-variance-skewness-kurtosis model with
cardinality constraint (MVSKC). Optimization for the MVSKC model is of great
difficulty in two parts. One is that the objective function is non-convex, the
other is the combinational nature of the cardinality constraint, leading to
non-convexity as well dis-continuity. Based on the observation that cardinality
constraint has the difference-of-convex (DC) property, we transform the
cardinality constraint into a penalty term and then propose three algorithms
including the proximal difference of convex algorithm (pDCA), pDCA with
extrapolation (pDCAe) and the successive convex approximation (SCA) to handle
the resulting penalized MVSK (PMVSK) formulation. Moreover, theoretical
convergence results of these algorithms are established respectively. Numerical
experiments on the real datasets demonstrate the superiority of our proposed
methods in obtaining high utility and sparse solutions as well as efficiency in
terms of time usage.
