In this paper we mainly focus on optimization of sums of squares of quadratic functions, which we refer to as second-order least-squares problems, subject to convex constraints. Our motivation arises from applications in risk parity portfolio selection. We generalize the setting further by considering a class of nonlinear, non convex functions which admit a (non separable) two-block representation with special structure. We then develop alternating direction and alternating linearization schemes for such functions and analyze their convergence and complexity. Due to the special structure of our functions, the steps of our methods reduce to solving convex optimization subproblems. We provide convergence rate results for the proposed methods. Furthermore, some global relaxation techniques are presented to find lower bounds and strengthen our local algorithms. We show the effectiveness of our techniques in application to risk parity optimization in portfolio management.
View Alternating direction methods for non convex optimization with applications to second-order least-squares and risk parity portfolio selection