TITLE: Finding a feasible point under quadratic constraints
QUESTION [5 upvotes]: Lets say we have a Quadratically Constrained Quadratic Program which we would like to optimize. The first step for many methods requires one to first find a point in the feasible region.
How can I answer the question "Is the feasible region non-empty?" If the answer is yes, I would like a witness obviously. I'm assuming this is possible in polynomial time if the problem is convex?

REPLY [5 votes]: For feasibility, only the constraints matter, not the objective. One way to identify a feasible point (even if there exist no strictly feasible point), is to solve the feasibility problem
$$
\min \ 0 \quad \text{subject to your constraints}.
$$
Usually, this is not a much better-behaved problem than the original problem. An alternative is to minimize the infeasibility residual
$$
\min \ \|\text{constraints}\|^2
$$
(assuming equality constraints). This is typically a nonlinear least-squares problem. Depending on the form of your constraints it can be solved more or less efficiently. If your original problem was feasible, this last one is a zero-residual least-squares problem and Gauss-Newton should converge quadratically.
That being said, note that many modern methods do not require an initial feasible point. I may be able to be more specific if you give more details on your problem.