Given an optimisation problem of the form , where , can we equivalently solve the problem ?
This question was published in Harvey J. Greenberg’s “Myths and Counterexamples in Mathematical Programming,” which can be found online (see NLP Myth 5). The author points out to:
D. M. Bloom. FFF #34. The shortest distance from a point to a parabola. The College Mathematics Journal, 22(2):131, 1991,
where Bloom addresses the simple problem of determining the shortest distance between the point on the plane and the parabola .
A simple sketch reveals that the unique minimiser of this problem is the point and the distance is equal to 5.
The original problem, using the squared distance, is
Now if we simply substitute by we get the problem
However, in doing so we have dropped the requirement that . This leads to the “paradox” that the solution of the second problem is which corresponds to an imaginary x-coordinate.
The correct way to go is, of course, to use the KKT conditions of the original problems to determine its critical points.