On optimization over the doubly nonnegative cone

Abstract

In recent years, many studies have focused on semidefinite relaxation for combinatorial optimization problems and their usefulness. While those studies force the solution matrix of the relaxation problem to be symmetric and positive semidefinite, we often see that each element of the solution matrix is meant to be nonnegative. A positive semidefinite matrix whose elements are nonnegative is called a doubly nonnegative matrix. It would be natural to obtain a better relaxation by considering an optimization problem over the set of such matrices (we call it the doubly nonnegative cone). In this paper, we will show that the doubly nonnegative relaxation gives significantly tight bounds for a class of quadratic assignment problems, while the computational time may not be affordable as long as we solve the relaxation as usual, i.e., as an optimization problem over the symmetric cone given by the direct sum of the semidefinite cone and the nonnegative orthant. Aiming to develop new and efficient algorithms for solving the doubly nonnegative optimization problems, we provide some basic properties of the doubly nonnegative cone focusing on barrier functions on its interior.