Algorithmic copositivity detection by simplicial partition

Copositive optimization – Recent developments and applications

“…This is in sharp contrast to the case of box-constrained QPs [6] and may be intuitively explained by the fact that, unlike the hypercube, the volume of the standard simplex decreases exponentially fast in n as dimension n increases. Also, this poses no contradiction to the well-known inapproximability results [72] for the clique number ω(G), since the StQP formulation (30) in Section 6.3 below provides the inverse value 1 ω(G) .…”

“…Again, we may employ branch-and-bound by discarding those ∆ i and subdivide others where such a sufficient criterion fails, until convergence or detection of a violating vector v ∈ R n + with v ⊤ Qv < 0. This is the common idea of the papers [30], [31], and [20].…”

“…In [30], the authors use subsimplices ∆ i of ∆, the sufficient condition Q i ∈ N , and in case of failure bisect the longest edge of ∆ i corresponding to a negative entry. A violating vector is detected whenever a negative diagonal element of Q i occurs (recall that these equal v ⊤ j Qv j for some vertex v j of ∆ i ).…”

Copositive Programming – a Survey

“…A conceptually different approach to copositivity testing which essentially uses global optimization techniques has been proposed in [18]. This approach relies on the observation that A is copositive iff the quadratic form x T Ax ≥ 0 on the standard simplex.…”

“…We also refer to [16] for a detailed elaboration. The method is based on the copositivity conditions developed in [18] which we briefly described in Section 4. Recall condition (6).…”