On computing the minima of quadratic forms (Preliminary Report)

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