SDP application on portfolio optimization problem with non-convex quadratic constraints

This paper is focused on the problem of portfolio optimization. This problem is formulated as a mean-risk model with a special choice of shortfall constraints that leads to quadratic programming. We show the methodology of finding an optimal solution to such problem even if the said quadratic programming is non-convex. The non-convexity comes from our demand that the model should favour the positive deviations from expected returns of the portfolio. First the former problem is converted to semi-definite programming (SDP) problem using semi-definite relaxation technique. In the literature it is shown that this will provide a good proximate for the former problem. The SDP is solved using interior point methods with self-concordant barrier. These methods have in worst case scenario polynomial time complexity. The last issue that we face is the issue of feasibility of the solution of the relaxed problem. We propose using eigenvalue method and randomization method to deal with this issue. The whole process is demonstrated on a real data. It is implemented in R using Rcsdp package.