Mixed SDP/SOCP moment relaxations of the optimal power flow problem

Recently, convex “moment” relaxations developed from the Lasserre hierarchy for polynomial optimization problems have been shown capable of globally solving many optimal power flow (OPF) problems. The moment relaxations, which take the form of semidefinite programs (SDP), generalize a previous SDP relaxation of the OPF problem. This paper presents an approach for improving the computational performance of the moment relaxations for many problems. This approach enforces second-order cone programming (SOCP) constraints that establish necessary (but not sufficient) conditions for satisfaction of the SDP constraints arising from the higher-order moment relaxations. The resulting “mixed SDP/SOCP” formulation implements the first-order relaxation using SDP constraints and the higher-order relaxations using SOCP constraints. Numerical results demonstrate that this mixed SDP/SOCP relaxation is capable of solving many problems for which the first-order moment relaxation fails to yield a global solution. For several examples, the mixed SDP/SOCP relaxation improves computational speed by factors from 1.13 to 18.7.