Exploiting Sparsity in SDP Relaxation for Harmonic Balance Method

In general, harmonic balance problems are extremely nonconvex and difficult to solve. A convex relaxation in the form of semidefinite programming has attracted a lot of attention recently, as it finds a global solution with high accuracy without the need for initial values. However, the computational cost of solving large-scale optimization poses a major challenge for the application in many real-world practical cases. This work proposes a heuristic optimization approach to find the Fourier coefficients of harmonic balance problems. The structural sparsity in the Harmonic Balance problem is exploited to improve numerical tractability and efficiency at the cost of adding smaller-sized semidefinite constraints in the problem formulation. After exploiting sparsity, the simulation results show that the size of the largest semidefinite constraint and the number of decision variables are greatly reduced. In addition, the computation speed shows an improvement rate of 3 or 7.5 times for larger instance problems and a reduction in memory occupation. Moreover, the proposed formulation can also solve nonlinear circuits with nonpolynomial nonlinearities with high accuracy.