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In this paper, we aim at solving the cardinality constrained high-orderportfolio optimization, i.e., mean-variance-skewness-kurtosis model withcardinality constraint (MVSKC). Optimization for the MVSKC model is of greatdifficulty in two parts. One is that the objective function is non-convex, theother is the combinational nature of the cardinality constraint, leading tonon-convexity as well dis-continuity. Based on the observation that cardinalityconstraint has the difference-of-convex (DC) property, we transform thecardinality constraint into a penalty term and then propose three algorithmsincluding the proximal difference of convex algorithm (pDCA), pDCA withextrapolation (pDCAe) and the successive convex approximation (SCA) to handlethe resulting penalized MVSK (PMVSK) formulation. Moreover, theoreticalconvergence results of these algorithms are established respectively. Numericalexperiments on the real datasets demonstrate the superiority of our proposedmethods in obtaining high utility and sparse solutions as well as efficiency interms of time usage.

Sparse High-Order Portfolios via Proximal DCA and SCA