ON THE NUMBER OF LOCAL MAXIMA OF A CONSTRAINED QUADRATIC FORM

We consider the problem of determining the greatest, number of local m axima that a quadratic form can have when the vector is constrained to lie within the unit simplex. Specifically, we investigate the local m axima of V = p(T)Ap, where P = (p1, p2, ... , p(n)) is-an-element-of D ELTA(n) {x is-an-element-of R(n): x(i) greater-than-or-equal-to 0, SIG MA(i)x(i) and A = (a(ij)) is a real. symmetric n x n matrix. Consideri ng the central role played by quadratic forms in the history of mathem atics in general and algebra in particular, it is perhaps surprising t hat this problem does not appear to have received any attention. It is a rather awkward problem because the constraint cannot be readily inc orporated. 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 incr eases geometrically with n and also present some results on the length s (i.e. the number of non-zero elements) of the maximizing vectors.