A multigrid approach to SDP relaxations of sparse polynomial optimization problems

Abstract

We propose a multigrid approach for the global optimization of polynomial optimization problems with sparse support. The problems we consider arise from the discretization of infinite dimensional optimization problems, such as PDE optimization problems, boundary value problems, and some global optimization applications. In many of these applications, the level of discretization can be used to obtain a hierarchy of optimization models that capture the underlying infinite dimensional problem at different degrees of fidelity. This approach, inspired by multigrid methods, has been successfully used for decades to solve large systems of linear equations. However, multigrid methods are difficult to apply to semidefinite programming (SDP) relaxations of polynomial optimization problems. The main difficulty is that the information between grids is lost when the original problem is approximated via an SDP relaxation. Despite the loss of information, we develop a multigrid approach and propose prolongation operators to relate the primal and dual variables of the SDP relaxation between lower and higher levels in the hierarchy of discretizations. We develop sufficient conditions for the operators to be useful in practice. Our conditions are easy to verify, and we discuss how they can be used to reduce the complexity of infeasible interior point methods. Our preliminary results highlight two promising advantages of following a multigrid approach compared to a pure interior point method: the percentage of problems that can be solved to a high accuracy is much greater, and the time necessary to find a solution can be reduced significantly, especially for large scale problems.