Effective algorithms for optimal portfolio deleveraging problem with cross impact

We investigate the optimal portfolio deleveraging (OPD) problem with permanent and temporary price impacts, where the objective is to maximize equity while meeting a prescribed debt/equity requirement. We take the real situation with cross impact among different assets into consideration. The resulting problem is, however, a nonconvex quadratic program with a quadratic constraint and a box constraint, which is known to be NP-hard. In this paper, we first develop a successive convex optimization (SCO) approach for solving the OPD problem and show that the SCO algorithm converges to a KKT point of its transformed problem. Second, we propose an effective global algorithm for the OPD problem, which integrates the SCO method, simple convex relaxation, and a branch-and-bound framework, to identify a global optimal solution to the OPD problem within a prespecified ε-tolerance. We establish the global convergence of our algorithm and estimate its complexity. We also conduct numerical experiments to demonstrate the effectiveness of our proposed algorithms with both real data and randomly generated medium- and large-scale OPD instances.