Doubly nonnegative relaxations are equivalent to completely positive reformulations of quadratic optimization problems with block-clique graph structures

We study the equivalence among a nonconvex QOP, its CPP and DNN relaxations under the assumption that the aggregate and correlative sparsity of the data matrices of the CPP relaxation is represented by a block-clique graph G. By exploiting the correlative sparsity, we decompose the CPP relaxation problem into a clique-tree structured family of smaller subproblems. Each subproblem is associated with a node of a clique tree of G. The optimal value can be obtained by applying an algorithm that we propose for solving the subproblems recursively from leaf nodes to the root node of the clique-tree. We establish the equivalence between the QOP and its DNN relaxation from the equivalence between the reduced family of subproblems and their DNN relaxations by applying the known results on: (1) CPP and DNN reformulation of a class of QOPs with linear equality, complementarity and binary constraints in 3 nonnegative variables. (2) DNN reformulation of a class of quadratically constrained convex QOPs with any size. (3) DNN reformulation of LPs with any size. As a result, we show that a QOP whose subproblems are the QOPs mentioned in (1), (2) and (3) is equivalent to its DNN relaxation, if the subproblems form a clique-tree structured family induced from a block-clique graph.