eldo_user

Figure 13-3. Illustration of Optimality Conditions
Optimization
Eldo Optimizer SQP Algorithm
Note
Previous definitions of the optimality conditions related only to bound constraint
minimization problems. The conditions OPTIM(x) have been extended to more general
problems that the Eldo optimizer SQP method can handle.
In many cases the function values are the result of extensive computation, possibly involving an
iterative procedure that can provide a few digits of precision at reasonable cost.
Suppose that a constraint function c (i) (x) is computed for some relevant x and the first six digits
are known to be correct. A constraint is considered as active at its upper bound (or lower bound)
if the difference between the values c (i) (x) and c (i) u (or c (i) l respectively) is less than some
tolerance of order 1.0 × 10 −6 .
This tolerance, δ, specifies how accurately the constraints are satisfied. It defines the maximum
absolute violation in non-linear constraints at a feasible point. A constraint is considered
satisfied if its violation does not exceed the tolerance δ.
The feasible region for the constraints c l (i) ≤ c (i) (x) ≤ c u (i) is shown in Figure 13-4.
Figure 13-4. Illustration of Constraints
The constraints are considered satisfied if c (i) (x) lies in the region 2, 3 or 4, and inactive if c (i) (x)
lies in region 3. The constraint c (i) l ≤ c (i) (x) is considered active in region 2, and violated in
region 1. Similarly, c (i) (x) ≤ c (i) u is active in region 4, and violated in region 5. For equality
constraints c
(i)
l = c (i) u , regions 2 and 4 are the same, and region 3 is empty. The default value is
appropriate when the constraints contain data about the accuracy.
Eldo® User's Manual, 15.3 625