An algorithm for determining the least minimum singular value of a polytope of matrices

Given m square matrices A₁, …, A_m, let A denote the set of all their convex combinations. Then the authors consider the problem of determining a member of A whose minimum singular value is the smallest. A related problem, known as robust nonsingularity problem, is to determine if every member of A is nonsingular. Clearly a solution to the authors´ problem automatically solves the robust nonsingularity problem. Unfortunately, the robust nonsingularity problem has been demonstrated to be NP-hard which in turn makes the authors´ problem NP-hard. To avoid this computational intractability, the authors provide an algorithm that computes a lower bound and an upper bound on the least minimum singular value within a prescribed tolerance. Of course, if the prescribed tolerance is set to zero then the authors´ algorithm would compute the least minimum singular value. The authors´ method makes use of the so-called simplicial algorithms