  We address the problems of minimizing and of maximizing the spectral radius
overa compact family of non-negative matrices. Those problems being hard in
generalcan be efficiently solved for some special families. We consider the
so-called prod-uct families, where each matrix is composed of rows chosen
independently from givensets. A recently introduced greedy method works very
fast. However, it is applicablemostly for strictly positive matrices. For
sparse matrices, it often diverges and gives awrong answer. We present the
"selective greedy method" thatworks equally well forall non-negative product
families, including sparse ones.For this method, we provea quadratic rate of
convergence and demonstrate its efficiency in numerical examples.The numerical
examples are realised for two cases: finite uncertainty sets and poly-hedral
uncertainty sets given by systems of linear inequalities. In dimensions up to
2000, the matrices with minimal/maximal spectral radii in product families are
foundwithin a few iterations. Applications to dynamical systemsand to the graph
theoryare considered
