Abstract: The portfolio selection model with factor risk control in the mean-variance framework is a nonconvex quadratically constrained quadratic programming problem that is known to be NP-hard. In this paper, we investigate the convex relaxation approach based on a non-redundant matrix splitting for the portfolio selection model with factor risk control. We first show that the convex relaxation based on a non-redundant matrix splitting can provide a stronger bound than a redundant one. A non-redundant matrix splitting is derived via solving an auxiliary SDP problem. We then present a new convex relaxation based on this non-redundant matrix splitting and analyze the properties of its optimal solutions and optimal values. We also show that the new relaxation model provides a stronger lower bound than one in the literature. Preliminary numerical results demonstrate that the branch-and-bound algorithm based on the new convex relaxation can find effectively the global optimal solution of the original problem.

Key words: Portfolio selection, mean variance, factor risk, convex relaxation, branch and bound