A new hierarchical divisive algorithm is proposed for identifying communities in complex networks. To that effect, the definition of community in the weak sense of Radicchi et al. is extended into a criterion for a bipartition to be optimal: one seeks to maximize the minimum for both classes of the bipartition of the ratio of inner edges to cut edges. A mathematical program is used within a dichotomous search to do this in an optimal way for each bipartition. This includes an exact solution of Wang et al.'s problem of detecting indivisible communities. The resulting hierarchical divisive algorithm is compared with exact modularity maximization on both artificial and real world datasets. For two problems of the former kind optimal solutions are found, for five problems of the latter kind the edge ratio algorithm always appears to be competitive. Moreover, it provides new information in several cases, notably through the use of the dendrogram summarizing the resolution. Finally, both algorithms are compared on reduced versions of the Girvan-Newman and of the Lancichinetti et al. datasets. Results for these instances appear to be comparable.