  Semi-infinite programs are a class of mathematical optimization problems with
a finite number of decision variables and infinite constraints. As shown by
Blankenship and Falk (Blankenship and Falk. "Infinitely constrained
optimization problems." Journal of Optimization Theory and Applications 19.2
(1976): 261-281.), a sequence of lower bounds which converges to the optimal
objective value may be obtained with specially constructed finite
approximations of the constraint set. In (Mitsos. "Global optimization of
semi-infinite programs via restriction of the right-hand side." Optimization
60.10-11 (2011): 1291-1308.), it is claimed that a modification of this lower
bounding method involving approximate solution of the lower-level program
yields convergent lower bounds. We show with a counterexample that this claim
is false, and discuss what kind of approximate solution of the lower-level
program is sufficient for correct behavior.
