A nonsmooth exclusion test for finding all solutions of nonlinear equations

Abstract

A new approach is proposed for finding all solutions of systems of nonlinear equations with bound constraints. The zero finding problem is converted to a global optimization problem whose global minima with zero objective value, if any, correspond to all solutions of the initial problem. A branch-and-bound algorithm is used with McCormick's nonsmooth convex relaxations to generate lower bounds. An inclusion relation between the solution set of the relaxed problem and that of the original non-convex problem is established which motivates a method to generate automatically reasonably close starting points for a local Newton-type method. A damped-Newton method with natural level functions employing the restrictive monotonicity test is employed to find solutions robustly and rapidly. Due to the special structure of the objective function, the solution of the convex lower bounding problem yields a nonsmooth root exclusion test which is found to perform better than earlier interval based exclusion tests. The Krawczyk operator based root inclusion and exclusion tests are also embedded in the proposed algorithm to refine the variable bounds for efficient fathoming of the search space. The performance of the algorithm on a variety of test problems from the literature is presented and for most of them the first solution is found at the first iteration of the algorithm due to the good starting point generation.