  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 non-convex 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 pre-specified $\epsilon$-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 the real data and the randomly generated medium- and
large-scale OPD problem instances.
