High-order Moment Portfolio Optimization via An Accelerated Difference-of-Convex Programming Approach and Sums-of-Squares

The Mean-Variance-Skewness-Kurtosis (MVSK) portfolio optimization model is a quartic nonconvex polynomial minimization problem over a polytope, which can be formulated as a Difference-of-Convex (DC) program. In this manuscript, we investigate four DC programming approaches for solving the MVSK model. First, two DC formulations based on the projective DC decomposition and the Difference-of-Convex-Sums-of-Squares (DC-SOS) decomposition are established, where the second one is novel. Then, DCA is applied to solve these DC formulations. The convergence analysis of DCA for the MVSK model is established. Second, we propose an accelerated DCA (Boosted-DCA) for solving a general convex constrained DC program involving both smooth and nonsmooth functions. The acceleration is realized by an inexact line search of the Armijo-type along the DC descent direction generated by two consecutive iterates of DCA. The convergence analysis of the Boosted-DCA is established. Numerical simulations of the proposed four DC algorithms on both synthetic and real portfolio datasets are reported. Comparisons with KNITRO, FILTERSD, IPOPT and MATLAB fmincon optimization solvers demonstrate good performance of our methods. Particularly, two DC algorithms with DC-SOS decomposition require less number of iterations, which demonstrates that DC-SOS decomposition can provide better convex over-approximations for polynomials. Moreover, the accelerated versions indeed reduce the number of iterations and achieve the best numerical results.