The economical way to use the Kuhn-Tucker theorem
M. Baltovic in his guide to Optimization Theory blindly checks the constraint qualification condition for all possible constraints combinations. This is not rational because a) the work can be substantially reduced by properly using the Kuhn-Tucker conditions and b) some constraint combinations may not have sense.
We illustrate this point using Example 6.4:
maximize subject to , , .
are assumed to be positive. (The solution can be found using also convexity).
The key is to eliminate simple cases one by one
Case 1. Are internal solutions possible? In case of internal solutions, the constraints are irrelevant and FOC's can be applied to the objective function alone. Equations give which is impossible.
Denote the boundaries The Kuhn-Tucker conditions are (the numbering follows that in Baltovic):
Remember that these conditions work only when the maximizing point satisfies the constraint qualification condition. It is obviously satisfied for considered separately.
Case 2. Suppose the maximum belongs to only (and not to or Then from (6.6) which is impossible.
Case 3. Similarly, if the maximum belongs to only then which is also impossible.
Case 4. If the maximum belongs only to then and from (6.3), (6.4) we see that From (6.5)-(6.7) then
This system can be easily solved to give the maximizing point
The value of the objective function at this point is
Case 5. The only possibilities that are left are , , Don't bother checking the constraint qualification for these points because a) it may fail, in which case the Kuhn-Tucker theorem does not apply, even though any of these points can be a maximum point, and b) even if it holds, none of these points may be a maximum point (the Kuhn-Tucker theorem provides just a necessary condition). Just check directly that at these points takes values lower than at
It's easy to see that
Case 6. The intersection is empty, so checking the constraint qualification condition for it is a waste of time.
The solution in Baltovic takes more than three pages.