On computing the minima of quadratic forms (Preliminary Report)

Abstract

The following problem was recently raised by C. William Gear [1]: Let F(x1,x2,...,xn) = &Sgr;i≤j a'ijxixj + &Sgr;i bixi +c be a quadratic form in n variables. We wish to compute the point x→(0) = (x1(0),...,xn(0)), at which F achieves its minimum, by a series of adaptive functional evaluations. It is clear that, by evaluating F(x→) at 1/2(n+1)(n+2)+1 points, we can determine the coefficients a'ij,bi,c and thereby find the point x→(0). Gear's question is, “How many evaluations are necessary?” In this paper, we shall prove that O(n2) evaluations are necessary in the worst case for any such algorithm.