Effective algorithms for separable nonconvex quadratic programming with one quadratic and box constraints

We consider in this paper a separable and nonconvex quadratic program (QP) with a quadratic constraint and a box constraint that arises from application in optimal portfolio deleveraging (OPD) in finance and is known to be NP-hard. We first propose an improved Lagrangian breakpoint search algorithm based on the secant approach (called ILBSSA) for this nonconvex QP, and show that it converges to either a subopti-mal solution or a global solution of the problem. We then develop a successive convex optimization (SCO) algorithm to improve the quality of suboptimal solutions derived from ILBSSA, and show that it converges to a KKT point of the problem. Second, we develop a new global algorithm (called ILBSSA-SCO-BB), which integrates the ILB-SSA and SCO methods, convex relaxation and branch-and-bound framework, to find a globally optimal solution to the underlying QP within a pre-specified e-tolerance. We establish the convergence of the ILBSSA-SCO-BB algorithm and its complex-ity. Preliminary numerical results are reported to demonstrate the effectiveness of the ILBSSA-SCO-BB algorithm in finding a globally optimal solution to large-scale OPD instances.