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

One of the techniques for solving systems of non-linear equations F1(x1,..., xn) = 0,..., Fn(x1,..., xn) = 0 (F ⃗ (x⃗) = 0⃗) is a homo-topy method, when we start with a solution of a simplified (and thus easier-to-solve) approximate systemGi(x1,..., xn) = 0, and then grad-ually adjust this solution by solving intermediate systems of equation Hi(x1,..., xn) = 0 for an appropriate “transition ” function H⃗(x⃗) = f⃗(λ, F ⃗ (x⃗), G⃗(x⃗)). The success of this method depends on the selec-tion of the appropriate combination function f⃗(λ, u⃗1, u⃗2). The most commonly used combination function is the convex homotopy function f⃗(λ, u⃗1, u⃗2) = λ·u⃗1+(1−λ)·u⃗2. In this paper, we provide a theoretical justification for this combination function. 1 Formulation of the Problem Need to solve systems of equations. In many practical situations, it is difficult (or even impossible) to directly measure the values of the desire