  The first moment and second central moments of the portfolio return, a.k.a.
mean and variance, have been widely employed to assess the expected profit and
risk of the portfolio. Investors pursue higher mean and lower variance when
designing the portfolios. The two moments can well describe the distribution of
the portfolio return when it follows the Gaussian distribution. However, the
real world distribution of assets return is usually asymmetric and
heavy-tailed, which is far from being a Gaussian distribution. The asymmetry
and the heavy-tailedness are characterized by the third and fourth central
moments, i.e., skewness and kurtosis, respectively. Higher skewness and lower
kurtosis are preferred to reduce the probability of extreme losses. However,
incorporating high-order moments in the portfolio design is very difficult due
to their non-convexity and rapidly increasing computational cost with the
dimension. In this paper, we propose a very efficient and convergence-provable
algorithm framework based on the successive convex approximation (SCA)
algorithm to solve high-order portfolios. The efficiency of the proposed
algorithm framework is demonstrated by the numerical experiments.
