Is my code correct for finding the distance between a point and a surface?

Hello

We have a point, an (hyper)surface and the distance function like . The surface in my example is .

P = [0.4 0.4 0.3];  % the point
f = @(x) sqrt(sum(x));  % the surface
distsq = @(x) sum((x-P(1:end-1)).^2) + (f(x)-P(end)).^2;  % the distance function squared,want to minimize
[x,fval] = fmincon(distsq,[0.5 0.5],[],[],[],[],[0 0],[1 1])

I want to go higher in dimensions and see how it performs. I just don't know how can I be somewhat sure that the result from fmincon is correct. I'm interested only in the hypercube.

Answer 1

[Matt J]

It's largely correct, except that your function distsq is not differentiable at x=0. So, if there's a chance the solution might lie there (but I think it's impossible if P(n) and at least one other P(i) are greater than zero), then I would make a transformation to get rid of the non-differentiability. In this case, this could be,

distsq = @(x) sum((x-P(1:end-1)).^2) + (f(x).^2-P(end).^2).^2;

Note however that for the specific f in your example, the transformation turns the problem into a linear least squares problem, so that you can use lsqlin instead of fmincon,

C=[speye(n-1);ones(1,n-1)];
d=P; d(end)=d(end)^2;

[x,fval] = lsqlin(C,d,[],[],[],[],[0 0],[1 1]);

This also has the advantage that lsqlin is globally convergent and doesn't require an initial guess.