Study of nonlinear power optimization problems using algebraic graph theory

This work is concerned with solving non-convex power optimization problems by introducing the concept of “nonlinear optimization over graph”. To this end, the structure of a given nonlinear real/complex optimization with quadratic arguments is mapped into a generalized weighted graph, where each edge is associated with a weight set constructed from the known parameters of the optimization (e.g., the coefficients). This generalized weighted graph captures both the sparsity of the optimization and possible patterns in the coefficients. The notion of “sign definite sets” is introduced for both real and complex weight sets, and it is then shown that the polynomial-time solvability of the optimization may be inferred from the topology of its associated graph together with the sign definiteness of its weight sets. As an application of this result, it is finally proved that a broad class of optimization problems over power networks are polynomial-time solvable via two convex relaxations due to the passivity of transmission lines.