Solving Polynomial Systems by Penetrating Gradient Algorithm Applying Deepest Descent Strategy

Abstract

An algorithm and associated strategy for solving polynomial systems within the optimization framework is presented. The algorithm and strategy are named, respectively, the penetrating gradient algorithm and the deepest descent strategy. The most prominent feature of penetrating gradient algorithm, after which it was named, is its ability to see and penetrate through the obstacles in error space along the line of search direction and to jump to the global minimizer in a single step. The ability to find the deepest point in an arbitrary direction, no matter how distant the point is and regardless of the relief of error space between the current and the best point, motivates movements in directions in which cost function can be maximally reduced, rather than in directions that seem to be the best locally (like, for instance, the steepest descent, i.e., negative gradient direction). Therefore, the strategy is named the deepest descent, in contrast but alluding to the steepest descent. Penetrating gradient algorithm is derived and its properties are proven mathematically, while features of the deepest descent strategy are shown by comparative simulations. Extensive benchmark tests confirm that the proposed algorithm and strategy jointly form an effective solver of polynomial systems. In addition, further theoretical considerations in Section 5 about solving linear systems by the proposed method reveal a surprising and interesting relation of proposed and Gauss-Seidel method.