squaring the equality constraints

Question

When creating an unconstrained optimization problem from an equality constrained one, the usual way to build the Lagrangian, is by adding a term consisting of a multiplier, multiplied by the equality constraint. Are there problem instances, where it makes better sense to square the equality constraint and then use that in the unconstrained problem?

Answer 1

When the equality constraint has the form $g(x)=0$ then replacing it with $h(x):=g^2(x)=0$ is fatal. Lagrange's method detects the points where $df(x)=\lambda dg(x)$ with a finite $\lambda$, and together with $g(x)=0$ you can expect to obtain a finite number of "conditionally stationary points" on the surface $S: \>g(x)=0$. But $dh(x)=0$ at all points of $S$, so the equation $df(x)=\lambda dh(x)$ will not produce a single point that is of interest.

It is another matter with the objective function $f$. When $f$ appears as a square root (e.g., if $f$ represents a distance between certain points) then no harm is done when you replace $f$ by $f^2$.