Spectral simplex method

We develop an iterative optimization method for nding the maximal and minimal spectral radius of a matrix over a compact set of nonnegative matrices. We consider matrix sets with product structure, i.e., all rows are chosen independently from given compact sets (row uncertainty sets). If all the uncertainty sets are nite or polyhedral, the algorithm nds the matrix with maximal/minimal spectral radius within a few iterations. It is proved that the algorithm avoids cycling and terminates within nite time. The proofs are based on spectral properties of rank-one corrections of nonnegative matrices. The practical eciency is demonstrated in numerical examples and statistics in dimensions up to 500. Some generalizations to non-polyhedral uncertainty sets, including Euclidean balls, are derived. Finally, we consider applications to spectral graph theory, mathematical economics, dynamical systems, and dierence equations.

• Type: Journal
• Year: 2016
• Where: MP
• Authors: Vladimir Yu. Protasov
• Keywords: Information Technology, MP 2016