Consider the constraint $$S_1 = \{(x, y) \; |\; \sqrt{x} + \sqrt{y} = 1 \}$$
How to use Lagrange Multipliers, when the constraint surface has a boundary?
In this case, after the Lagrange multiplier method gives candidates for maxima/minima, we need to check the "boundary points" of $S_1$, namely, $(1,0)$ and $(0,1)$ to get the global max/min. I can see that these two are "boundary points" intuitively when I plot the curve.
However, instead if the constraint set be
$$S_2 = \{ (x, y) \; |\; x^2 + y^2 = 1\},$$ then in this question, one answer states that for this constraint set, there is no "boundary point". Constrained Extrema: How to find end points of multivariable functions for global extrema
The only difference I see is that pictorially, one is a closed curve, but the other is not.
However, I am unable to see what is the mathematical definition that will allow me to conclude that $S_1$ has boundary points $(0, 1)$ and $(1,0)$ and $S_2$ has none?
Q) What is the definition of "end point" or "boundary point" being used here that explains both $S_1$, $S_2$.
Best Answer
In many extremal problems the set $S\subset{\mathbb R}^n$ on which the extrema of some function $f$ are sought is stratified, i.e., consists of points of different nature: interior points, surface points, edges, vertices. If an extremum is assumed in an interior point it comes to the fore as solution of the equation $\nabla f(x)=0$. An extremum which is at a (relative) interior point of a surface or an edge comes to the fore by Lagrange's method or via a parametrization of this surface or edge. Here (relative) interior refers to the following: Lagrange's method deals only with constrained points from which you can march in all tangent directions of the submanifold (surface, edge, $\ldots$) defined by the constraint(s), all the while remaining in $S$. Now at a vertex there are forbidden marching directions on all surfaces meeting at that vertex. If the extremum is taken on such a vertex it only comes to the fore if you have deliberately taken all vertices into your candidate list.
Now your $S_1$ is an arc in the plane with two endpoints. (The latter are not immediately visible in your presentation of $S_1$, but you have found them.) Your candidate list then should contain all relative interior points of the arc delivered by Lagrange's method plus the two boundary points.
The circle $S_2\!: \ x^2+y^2=1$ however has only "interior" points. The candidate list then contains only the points found by Lagrange's method.