  Generalized Frobenius powers of an ideal were introduced in work of
Hern\'andez, Teixeira, and Witt as characteristic-dependent analogs of test
ideals. However, little is known about the Frobenius powers and critical
exponents of specific ideals, even in the monomial case. We describe an
algorithm to compute the critical exponents of monomial ideals and use this
algorithm to prove some results about their Frobenius powers and critical
exponents. Rather than using test ideals, our algorithm uses techniques from
linear optimization.
