Inverse function theorem for polynomial equations using semidefinite programming

This paper is concerned with obtaining the inverse of polynomial functions using semidefinite programming (SDP). Given a polynomial function and a nominal point at which the Jacobian of the function is invertible, the inverse function theorem states that the inverse of the polynomial function exists at a neighborhood of the nominal point. In this work, we show that this inverse function can be found locally using convex optimization. More precisely, we propose infinitely many SDPs, each of which finds the inverse function at a neighborhood of the nominal point. We also design a convex optimization to check the existence of an SDP problem that finds the inverse of the polynomial function at multiple nominal points and a neighborhood around each point. This makes it possible to identify an SDP problem (if any) that finds the inverse function over a large region. As an application, any system of polynomial equations can be solved by means of the proposed SDP problem whenever an approximate solution is available. The method developed in this work is numerically compared with Newton's method and the nuclear-norm technique.