In this paper, we consider minimizing the ratio of two indefinite quadratic functions subject to two quadratic constraints. Using the extension of Charnes– Cooper transformation, we transform the problem to a homogenized quadratic problem. Then, we show that, under certain assumptions, it can be solved to global optimality using semidefinite optimization relaxation.

Laurent and Poljak introduced a very general class of valid linear inequalities, called gap inequalities, for the max-cut problem. We show that an analogous class of inequalities can be defined for general nonconvex mixed-integer quadratic programs. These inequalities dominate some inequalities arising from a natural semidefinite relaxation.

The problem of colouring a k-colourable graph is well-known to be NP-complete, for k ≥ 3. The MAX-k-CUT approach to approximate k-colouring is to assign k colours to all of the vertices in polynomial time such that the fraction of ‘defect edges’ (with endpoints of the same colour) is provably small. The best known approximation was obtained by Frieze and Jerrum [9], using a semidefinite program...

The Goemans-Williamson randomized algorithm guarantees a high-quality approximation to the Max-Cut problem, but the cost associated with such an approximation can be excessively high for large-scale problems due to the need for solving an expensive semidefinite relaxation. In order to achieve better practical performance, we propose an alternative, rank-two relaxation and develop a specialized ...