It has been long-conjectured that the crossing number cr(K m,n ) of the complete bipartite graph K m,n equals the Zarankiewicz Number Z(m, n) :Another long-standing conjecture states that the crossing number cr(K n ) of the complete graph K n equals Z(n) := 1 4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values:The previous best known lower bounds were 0.8m/(m − 1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K 7,n ) ≥ 2.1796n 2 − 4.5n. To obtain this improved lower bound for cr(K 7,n ), we use some elementary topological facts on drawings of K 2,7 to set up a quadratic program on 6! variables whose minimum p satisfies cr(K 7,n ) ≥ (p/2)n 2 − 4.5n, and then use state-ofthe-art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p ≥ 4.3593.