On a generalized Collatz-Wielandt formula and finding saddle-node bifurcations

We introduce the nonlinear generalized Collatz-Wielandt formula $$\lambda^*= \sup_{x\in Q}\min_{i:h_i(x) \neq 0} \frac{g_i(x)}{ h_i(x)}, ~~Q \subset \mathbb{R}^n,$$ and prove that its solution $(x^*,\lambda^*)$ yields the maximal saddle-node bifurcation for systems of equations of the form: $g(x)-\lambda h(x)=0, ~~x \in Q$. Using this we introduce a simply verifiable criterion for the detection of saddle-node bifurcations of a given system of equations. We apply this criterion to prove the existence of the maximal saddle-node bifurcations for finite-difference approximations of nonlinear partial differential equations and for the system of power flow equations.