Homotopy Techniques in Solving Systems of Nonlinear Equations: A Theoretical Justification of Convex Combinations

One of the techniques for solving systems of non-linear equations F1(x1,...,xn) = 0, ..., Fn(x1,...,xn) = 0, (F(x) = 0 in vector notations) is a homotopy method, when we start with a solution of a simplified (and thus easier-to-solve) approximate system Gi(x1,...,xn) = 0, and then gradually adjust this solution by solving intermediate systems of equation Hi(x1,...,xn) = 0 for an appropriate transition function H(x) = f(λ,F(x),G(x)). The success of this method depends on the selection of the appropriate combination function f(λ,u1,u2). The most commonly used combination function is the convex homotopy function f(λ,u1,u2) = λ * u1 + (1 − λ) * u2. In this paper, we provide a theoretical justification for this combination function