  This paper presents a convex sufficient condition for solving a system of
nonlinear equations under parametric changes and proposes a sequential convex
optimization method for solving robust optimization problems with nonlinear
equality constraints. By bounding the nonlinearity with concave envelopes and
using Brouwer's fixed point theorem, the sufficient condition is expressed in
terms of closed-form convex inequality constraints. We extend the result to
provide a convex sufficient condition for feasibility under bounded
uncertainty. Using these conditions, a non-convex optimization problem can be
solved as a sequence of convex optimization problems, with feasibility and
robustness guarantees. We present a detailed analysis of the performance and
complexity of the proposed condition. The examples in polynomial optimization
and nonlinear network are provided to illustrate the proposed method.
