project a point onto the intersection of surfaces

Question

I have several non linear equations $g_i$ that represent surfaces $s_i$. Their intersection form the surface $S$.

For example

$s_1 : g_1(x_1,x_2,...,x_n)=c_1$

...

$s_n : g_m(x_1,x_2,...,x_n)=c_m$

$S=s_1 \cap s_2 \cap ... \cap g_n$

Now I want to project a point $P$ onto $S$. I wish to find and algorithmic procedure. If one knows the surface $S$ equation it is trivial, because we just need to follow the gradient until we reach the right level set. But for this case I don't find.

It maybe looks a bit like this problem

Note: I don't want to use some analytic approach using linear algebra that I can find in optimization textbook, because I am going to deal with a huge number of surfaces, so I will never be able to solve the matrices.

Thank you

Answer 1

The projection is defined as the point $X=(x_1,x2,\ldots,x_n)$ such that solves

$\min || P - X ||^2$

$s.t.$

$ g_i(X) = c \quad \forall i=1,\ldots,m$

which can be trivial or extremely difficult depending on the constraint set. The solution is not even unique in general. If $g_i$'s are convex functions, than you can somehow solve efficiently. Otherwise you end up in a global optimization problem.