The complex zeros of a general complex polynomial are localized by constructing the intersection of areas in the complex plane defined by various inequality bounds on the eigenvalues of the companion matrix and also, possibly, by other inequalities on the zeros of polynomials. This localization then provides an efficient starting point for determining the zeros by applying a non-linear optimizer, such as the Fletcher-Powell method, to the square of the modulus of the polynomial, |p(x+iy)|2, in order to determine its minimums. The minimums of | p |2are zero and occur at the zeros of p(z). Experimentation indicates that Gershgorin's discs and similar results for Cassini's ovals supply rather sharp bounds for this purpose