Verification of feasibility proceeds as in [6], as also described in [8, §5.2.4,].
The upper bound of an interval value of the objective function
, where 
is a point or small interval vector,
represents an upper bound on the global optimum only if is
known to contain feasible points of
Problem 2. To obtain such verified feasible
points, the approximate feasible point finder mentioned in
§10 is first applied, then
a small box is constructed about this feasible point.
Certain coordinates of are selectively fixed at their
midpoints, and an interval Newton method as in §6 is
applied in the subspace defined by the coordinates that are not
fixed. See the figure in [7] or
[8, Fig. 5.3,].
Because feasibility must be verified to obtain an upper bound on the objective function, adding equality constraints to an unconstrained problem can result in longer running times in find_global_min.
As with verification of existence and uniqueness of critical points, the tolerance in the approximate feasible point finder is set smaller than the widths of the small constructed box, so that actual feasible points will lie near the center of the constructed box. This increases the chances that the interval Newton method will succeed in verification.
If an inequality constraint is certainly feasible, that is, if the
interval extension 
, then 
is ignored in verifying
feasibility. Otherwise, is treated as an equality constraint
in the verification process.