Robust Feasibility of Systems of Quadratic Equations Using Topological Degree Theory

We consider the problem of measuring the margin of robust feasibility ofsolutions to a system of nonlinear equations. We study the special case of asystem of quadratic equations, which shows up in many practical applicationssuch as the power grid and other infrastructure networks. This problem is ageneralization of quadratically constrained quadratic programming (QCQP), whichis NP-Hard in the general setting. We develop approaches based on topologicaldegree theory to estimate bounds on the robustness margin of such systems. Ourmethods use tools from convex analysis and optimization theory to cast theproblems of checking the conditions for robust feasibility as a nonlinearoptimization problem. We then develop inner bound and outer bound proceduresfor this optimization problem, which could be solved efficiently to derivelower and upper bounds, respectively, for the margin of robust feasibility. Weevaluate our approach numerically on standard instances taken from the MATPOWERand NESTA databases of AC power flow equations that describe the steady stateof the power grid. The results demonstrate that our approach can produce tightlower and upper bounds on the margin of robust feasibility for such instances.