Finding all solutions of systems of nonlinear equations with free variables

Systems of nonlinear equations often represent mathematical models in engineering design. This study proposes a novel method for finding all solutions of systems of nonlinear equations with free variables. The original problem is first transformed into a global optimization problem whose multiple global minima with a zero objective value correspond to all solutions of the original problem. Then, by using variable substitution on free variables and applying convexification strategies and piecewise linearization techniques on nonconvex functions, the transformed optimization problem is reformulated as a convex mixed-integer program solvable to reach an approximately global optimum. An algorithm is developed to find all solutions of the reformulated problem. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method in engineering design.