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

摘要

One of the techniques for solving systems of non-linear equations F_1(x_1,…,x_n) = 0, ..., F_n(x_1,… ,x_n) = 0 (F(x) = 0) is a homotopy method, when we start with a solution of a simplified (and thus easier-to-solve) approximate system G_i(x_1,… ,x_n) = 0, and then gradually adjust this solution by solving intermediate systems of equation H_i(x_1, … ,x_n) = 0 for an appropriate "transition" function H(x) = f(λ,F,G(x),G(x)). The success of this method depends on the selection of the appropriate combination function f(λ,u_1,u_2). The most commonly used combination function is the convex homotopy function f(λ, u_1,u_2) = λ · u_1 + (1-λ) · u_2. In this paper, we provide a theoretical justification for this combination function.