1 research outputs found
Solving Polynomial Systems by Penetrating Gradient Algorithm Applying Deepest Descent Strategy
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