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We address the problems of minimizing and of maximizing the spectral radiusovera compact family of non-negative matrices. Those problems being hard ingeneralcan be efficiently solved for some special families. We consider theso-called prod-uct families, where each matrix is composed of rows chosenindependently from givensets. A recently introduced greedy method works veryfast. However, it is applicablemostly for strictly positive matrices. Forsparse matrices, it often diverges and gives awrong answer. We present the"selective greedy method" thatworks equally well forall non-negative productfamilies, including sparse ones.For this method, we provea quadratic rate ofconvergence and demonstrate its efficiency in numerical examples.The numericalexamples are realised for two cases: finite uncertainty sets and poly-hedraluncertainty sets given by systems of linear inequalities. In dimensions up to2000, the matrices with minimal/maximal spectral radii in product families arefoundwithin a few iterations. Applications to dynamical systemsand to the graphtheoryare considered

The greedy strategy in optimizing the Perron eigenvalue