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

We introduce the nonlinear generalized Collatz-Wielandt formula $${\lambda }^{\ast }=\underset{x\in Q}{\sup }\underset{i:h_i(x)\mathrm{\ne }0}{\min }\frac{g_i(x)}{h_i(x)},Q\subset {\mathbb{R}}^n,$$ and prove that its solution $(x^{\ast },{\lambda }^{\ast })$ 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.