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.
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