  Solving large-scale systems of nonlinear equations/inequalities is a
fundamental problem in computing and optimization. In this paper, we propose a
generic successive projection (SP) framework for this problem. The SP
sequentially projects the current iterate onto the constraint set corresponding
to each nonlinear (in)equality. It extends von Neumann's alternating projection
for finding a point in the intersection of two linear subspaces, Bregman's
method for finding a common point of convex sets and the Kaczmarz method for
solving systems of linear equations to the more general case of multiple
nonlinear and nonconvex sets. The existing convergence analyses on randomized
Kaczmarz are merely applicable to linear case. There are no theoretical
convergence results of the SP for solving nonlinear equations. This paper
presents the first proof that the SP locally converges to a solution of
nonlinear equations/inequalities at a linear rate. Our work establishes the
convergence theory of the SP for the case of multiple nonlinear and nonconvex
sets. Besides cyclic and randomized projections, we devise two new greedy
projection approaches that significantly accelerate the convergence.
Furthermore, the theoretical bounds of the convergence rates are derived. We
reveal that the convergence rates are related to the Hoffman constants of the
Jacobian matrix of the nonlinear functions at the solution. Applying the SP to
solve the graph realization problem, which attracts much attention in
theoretical computer science, is discussed.
