Higher-order Moment Portfolio Optimization via The Difference-of-Convex Programming and Sums-of-Squares

Abstract

We are interested in developing a Difference-of-Convex (DC) programming approach based on Difference-of-Convex-Sums-of-Squares (DC-SOS) decomposition techniques for high-order moment (Mean-Variance-Skewness-Kurtosis) portfolio optimization model. This problem can be formulated as a nonconvex quartic multivariate polynomial optimization, then a DC programming formulation based on the recently developed DC-SOS decomposition is investigated. We can use a well-known DC algorithm, namely DCA, for its numerical solution. Moreover, an acceleration technique for DCA, namely Boosted-DCA (BDCA), based on an inexact line search (Armijo-type line search) to accelerate the convergence of DCA for smooth and nonsmooth DC program with convex constraints is proposed. This technique is applied to DCA based on DC-SOS decomposition, and DCA based on universal DC decomposition. Numerical simulations of DCA and Boosted-DCA on synthetic and real datasets are reported. Comparisons with some non-dc programming based optimization solvers (KNITRO, FILTERSD, IPOPT and MATLAB fmincon) demonstrate that our Boosted-DC algorithms can achieve same numerical results with good performance comparable to these efficient methods on solving the high-order moment portfolio optimization model.Comment: 42 pages, 13 figure