Combining SOS and Moment Relaxations with Branch and Bound to Extract Solutions to Global Polynomial Optimization Problems

In this paper, we present a branch and bound algorithm for extracting approximate solutions to Global Polynomial Optimization (GPO) problems with bounded feasible sets. The algorithm is based on a combination of SOS/Moment relaxations and successively bisecting a hyper-rectangle containing the feasible set of the GPO problem. At each iteration, the algorithm makes a comparison between the volume of the hyper-rectangles and their associated lower bounds to the GPO problem obtained by SOS/Moment relaxations to choose and subdivide an existing hyper-rectangle. For any desired accuracy, if we use sufficiently large order of SOS/Moment relaxations, then the algorithm is guaranteed to return a suboptimal point in a certain sense. For a fixed order of SOS/Moment relaxations, the complexity of the algorithm is linear in the number of iterations and polynomial in the number of constrains. We illustrate the effectiveness of the algorithm for a 6-variable, 5-constraint GPO problem for which the ideal generated by the equality constraints is not zero-dimensional - a case where the existing Moment-based approach for extracting the global minimizer might fail.