  We consider the problem of measuring the margin of robust feasibility of
solutions to a system of nonlinear equations. We study the special case of a
system of quadratic equations, which shows up in many practical applications
such as the power grid and other infrastructure networks. This problem is a
generalization of quadratically constrained quadratic programming (QCQP), which
is NP-Hard in the general setting. We develop approaches based on topological
degree theory to estimate bounds on the robustness margin of such systems. Our
methods use tools from convex analysis and optimization theory to cast the
problems of checking the conditions for robust feasibility as a nonlinear
optimization problem. We then develop inner bound and outer bound procedures
for this optimization problem, which could be solved efficiently to derive
lower and upper bounds, respectively, for the margin of robust feasibility. We
evaluate our approach numerically on standard instances taken from the MATPOWER
and NESTA databases of AC power flow equations that describe the steady state
of the power grid. The results demonstrate that our approach can produce tight
lower and upper bounds on the margin of robust feasibility for such instances.
