Trajectory-Based Methods for Indefinite Quadratic Programming Via Dynamical Systems

Abstract

This paper presents a comprehensive study on solving indefinite quadratic programming problems through the framework of dynamical systems. Specifically, we construct and analyze dynamical systems associated with two core DC (Difference of Convex functions) programming algorithms tailored for indefinite quadratic programs. Leveraging tools from non-smooth analysis and the theory of ordinary differential equations, we rigorously establish the existence and uniqueness of global solutions to these systems. Furthermore, we prove the convergence of system trajectories to Karush-Kuhn-Tucker (KKT) points, offering a novel trajectory-based verification method for KKT optimality conditions. The proposed approach not only provides theoretical insights into the behavior of such systems but also contributes to the development of robust algorithms for non-convex optimization problems.