The real structured singular value is hardly approximable

This paper investigates the problem of approximating the real structured singular value (real /spl mu/). A negative result is provided which shows that the problem of checking if /spl mu/=0 is NP-hard. This result is much more negative than the known NP-hard result for the problem of checking if /spl mu/<1. One implication of our result is that /spl mu/ is hardly approximable in the following sense: there does not exist an algorithm, polynomial in the size n of the /spl mu/ problem, which can produce an upper bound /spl mu/ for /spl mu/ with the guarantee that /spl mu//spl les//spl mu//spl les/K(n)/spl mu/ for any K(n)>0 (even exponential functions of n), unless NP=P. A similar statement holds for the lower bound of /spl mu/. Our result strengthens a recent result by Toker, which demonstrates that obtaining a sublinear approximation for /spl mu/ is NP-hard.