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

## TL;DR

This paper demonstrates that for quadratic optimization problems with block-clique graph structures, the nonnegative and completely positive relaxations are equivalent, enabling efficient decomposition and solution via clique-tree algorithms.

## Contribution

It establishes the equivalence between quadratic optimization problems and their DNN relaxations under block-clique graph sparsity, using a novel recursive decomposition approach.

## Key findings

- DNN relaxations are equivalent to CPP reformulations for block-clique structured QOPs.
- Decomposition into clique-tree structured subproblems facilitates solving large QOPs.
- The approach applies known reformulations to establish equivalence in specific subproblem classes.

## Abstract

We study the equivalence among a nonconvex QOP, its CPP and DNN relaxations under the assumption that the aggregated 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: (i) CPP and DNN reformulation of a class of QOPs with linear equality, complementarity and binary constraints in 4 nonnegative variables. (ii) DNN reformulation of a class of quadratically constrained convex QOPs with any size. (iii) DNN reformulation of LPs with any size. As a result, we show that a QOP whose subproblems are the QOPs mentioned in (i), (ii) and (iii) is equivalent to its DNN relaxation, if the subproblems form a clique-tree structured family induced from a block-clique graph.