The problem of minimizing a quadratic form over the unit simplex, referred to as a standard quadratic optimization problem, admits an exact reformulation as a linear optimization problem over the convex cone of completely positive matrices. This computationally intractable cone can be approximated from the inside and from the outside by two sequences of nested polyhedral cones of increasing accuracy. We investigate the sequences of upper and lower bounds on the optimal value of a standard quadratic optimization problem arising from these two hierarchies of inner and outer polyhedral approximations. We give a complete description of the structural properties of the instances on which upper and lower bounds are exact at a finite level of the hierarchy, and the instances on which upper and lower bounds converge to the optimal value only in the limit.
Technical Report, Department of Industrial Engineering, Koc University, Sariyer, Istanbul, Turkey.