On the number of local maxima of a constrained quadratic form

We consider the problem of determining the greatest number of local maxima that a quadratic form can have when the vector is constrained to lie within the unit simplex. Specifically, we investigate the local maxima of V = pTAp, where p = (p1, p2, . . . , pn)∊ ∆n = {X ∊ Rn: xi ≽ 0, ∑ixi = 1} and A = (aij) is a real, symmetric n x n matrix. Considering the central role played by quadratic forms in the history of mathematics in general and algebra in particular, it is perhaps surprising that this problem does not appear to have received any attention. It is a rather awkward problem because the constraint cannot be readily incorporated. A complete solution to the problem is lacking, but we show that the greatest number of maxima that any n x n matrix can have increases geometrically with n and also present some results on the lengths (i. e. the number of non-zero elements) of the maximizing vectors.