Solution of nonlinear system of equations through homotopy path

TL;DR

This paper introduces a homotopy method with a vector parameter to solve nonlinear complementarity problems by transforming them into systems of nonlinear equations, demonstrating effectiveness through an oligopolistic market equilibrium example.

Contribution

It proposes a novel homotopy continuation method with a vector parameter for solving nonlinear complementarity problems via nonlinear equations, ensuring a smooth and bounded solution path.

Findings

Effective solution of nonlinear complementarity problems demonstrated

Homotopy path remains smooth and bounded under certain conditions

Application to oligopolistic market equilibrium confirms practical utility

Abstract

The paper aims to show the equivalency between nonlinear complementarity problem and the system of nonlinear equations. We propose a homotopy method with vector parameter $\lambda$ in finding the solution of nonlinear complementarity problem through a system of nonlinear equations. We propose a smooth and bounded homotopy path to obtain solution of the system of nonlinear equations under some conditions. An oligopolistic market equilibrium problem is considered to show the effectiveness of the proposed homotopy continuation method. Keywords: Nonlinear complementarity problem, system of nonlinear equations, homotopy function with vector parameter, bounded smooth curve, oligopolistic market equilibrium.