On the exactness of semidefinite relaxation for nonlinear optimization over graphs: Part I

This work is concerned with finding a global optimization technique for a broad class of nonlinear optimization problems, including quadratic and polynomial optimizations. The main objective of this two-part paper is to investigate how the (hidden) structure of a given real/complex-valued optimization makes the problem easy to solve. To this end, three conic relaxations are proposed. Necessary and sufficient conditions are derived for the exactness of each of these relaxations, and it is shown that these conditions are satisfied if the optimization is highly structured. More precisely, the structure of the optimization is mapped into a generalized weighted graph, where each edge is associated with a weight set extracted from the coefficients of the optimization. In the real-valued case, it is shown that the relaxations are all exact if each weight set is sign definite and in addition a condition is satisfied for each cycle of the graph. It is also proved that if some of these conditions are violated, the relaxations still provide a low-rank solution for weakly cyclic graphs. In the complex-valued case, the notion of “sign definite complex sets” is introduced for complex weight sets. It is then shown that the relaxations are exact if each weight set is sign definite (with respect to complex numbers) and the graph is acyclic. In part II of the paper, the complex-valued case is further studied for cyclic graphs and moreover the application of this two-part paper in power system is thoroughly discussed.