While in optimization problems, usually, the goal is to find an extremum, I'm interested in finding the first feasible solution that satisfies all constraints (i.e. the nearest critical point to my initial point). In other words, I want to move from the initial value of the objective function and stop exploring as soon as the constraints are satisfied. Is there any safe technique to find exactly the nearest solution?
To clarify, see this image:
I'm currently using gradient descent to find the critical points of the Lagrangian formed from the objective function and the constraints. The critical points of Lagrangians occur at saddle points, rather than at local maxima (or minima). Unfortunately, many numerical optimization techniques (e.g. gradient descent) are designed to find local maxima (or minima) and not saddle points. So, in my problem, it moves over (or escapes) from some critical points in its path and tends to find an extremum.