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 in various ways from the inside and from the outside by two sequences of nested tractable convex cones of increasing accuracy. In this paper, we focus on the inner polyhedral approximations due to Yıldırım (Optim Methods Softw 27(1):155–173, 2012) and the outer polyhedral approximations due to de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, 2002). 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 complete algebraic descriptions of the sets of instances on which upper and lower bounds are exact at any given finite level of the hierarchy. We identify the structural properties of the sets of instances on which upper and lower bounds converge to the optimal value only in the limit. We present several geometric and topological properties of these sets. Our results shed light on the strengths and limitations of these inner and outer polyhedral approximations in the context of standard quadratic optimization.
- Completely positive cone
- Copositive cone
- Polyhedral approximations
- Standard quadratic optimization