A multilevel parallelized branch and bound algorithm for quadratic optimization

General QOPs (quadratic optimization problems) have a linear objective function cTx to be maximized over a nonconvex compact feasible region F described by a finite number of quadratic inequalities. Difficulties in solving a QOP arise from the nonconvexity in its quadratic terms. We propose a branch and bound algorithm for QOPs where branching operations have been designed to effectively reduce the nonconvexity of a given QOP so that the sub-QOPs generated during branching operations become easier to solve. The bounding procedure employed in our branch and bound algorithm is a successive convex relaxation algorithm based on semidefinite programming. A significant number of semidefinite programming problems involved in the algorithm are solved in parallel using the message passing interface library for message passing. This parallel implementation enables us to solve some highly nonconvex QOPs. Message passing and multithreading are mixed to improve the performance and parallel efficiency.