[For Part I see the preceding review.] An algorithm is presented to find a globally \(\varepsilon\)-optimal value of \(f\) (a Lipschitz function on \([a, b]\)), and a corresponding point. The algorithm is in two phases. In the first phase, the algorithm rapidly obtains a solution which is often globally \(\varepsilon\)-optimal. In the second phase, the algorithm either proves the \(\varepsilon\)-optimality of this solution, or finds a sequence of points containing one with a globally \(\varepsilon\)-optimal value. The algorithm is compared with existing algorithms, and performs favourably. In the second half of the paper a modification of the Piyavskii-Shubert algorithm is given to find a set of disjoint subintervals of \([a, b]\), containing only points with a globally \(\varepsilon\)-optimal value, such that the union contains all globally optimal points.