Essentially I am trying to develop an algorithm to minimize a function over a convex set that I don't know explicitly. However, I have a starting point "deepest in the set" (i.e. with largest norm distance to the boundaries), and a way to check if the feasibility criterion, $g(x) \le 0$, is met: I can call the function and check.
Here is a picture of my set, which is convex. Unfortunately I only obtain it numerically, hence not describing by functions. The red part is where my constraint is true, and X is my starting point, right in the "middle":
So I start my algorithm at x = X, a feasible starting point. Assuming the rest of my problem is convex, my plan is to take Newton steps from X and use a backtracking line search to determine step size while checking that I'm still feasible with respect to $g(x) \le 0$.
Is it the case that when I can no longer take a step size $t > \epsilon$ because I won't be feasible (i.e. $g(x) \le 0$ but $g(x + tv) \gt 0$ for all $t > \epsilon$), that I'll be at a point on the boundary, and thus optimal?
Extending on that, is this a valid iteration approach in general if your iterative algorithm can start at a point "deepest" in the convex set?