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.

💡 Research Summary

The paper introduces a novel approach for solving systems of polynomial equations by recasting the problem as an unconstrained optimization task and then applying a new line‑search technique called the Penetrating Gradient algorithm together with a meta‑strategy named Deepest Descent.
The authors begin by defining the residual vector r(x) with components r_i(x)=p_i(x) for each polynomial p_i, and they form the scalar objective F(x)=∑ r_i(x)^2. Traditional line‑search methods approximate the one‑dimensional function f(t)=F(x_k+t d) or use iterative bracketing, which can become trapped in local minima and require many function evaluations.
The Penetrating Gradient algorithm departs from this paradigm by exploiting the fact that f(t) is itself a polynomial when the search direction d is fixed. By expanding each residual p_i(x_k+t d) symbolically, squaring, and summing, the authors obtain an explicit polynomial g(t) of degree at most 2 · deg_max. The derivative g′(t) is also a polynomial, and its real roots are computed exactly (using Sturm sequences, Descartes’ rule of signs, or robust root‑finding methods such as Laguerre’s algorithm). Among all real stationary points, the algorithm selects the one that yields the smallest value of g(t). The new iterate is then x_{k+1}=x_k+t* d, which “penetrates” any intervening hills or valleys in the error landscape and lands directly at the global minimizer along that line. Because the line search is exact, the method does not suffer from step‑size restrictions or backtracking, and it can make arbitrarily large jumps when the objective permits.
The Deepest Descent strategy builds on this capability by evaluating the penetrating gradient along several candidate directions in each iteration (e.g., the negative gradient, coordinate axes, random unit vectors, and directions that performed well in previous steps). The direction that yields the greatest reduction in F is chosen for the next update. This contrasts with the classic steepest‑descent rule, which always follows the local gradient and can be misled by narrow valleys. By deliberately seeking the “deepest” descent direction, the method often avoids the slow zig‑zag behavior typical of gradient‑based schemes.
Mathematical analysis in the paper proves that the exact line search guarantees that the selected t* is a global minimizer of g(t) and that the algorithm converges to a stationary point of F under mild smoothness assumptions. The authors also discuss computational complexity: expanding the polynomial costs O(m·deg_max^2) operations, and root‑finding is polynomial in the degree, making the method practical for moderate‑degree systems. For very high degrees, they suggest degree‑reduction techniques such as monomial ordering or truncation to keep the cost manageable.
A particularly interesting theoretical observation is made for linear systems A x = b. When the objective F(x)=‖A x − b‖^2 is minimized using the penetrating gradient with coordinate‑axis directions, the update formula reduces exactly to the Gauss‑Seidel iteration. This reveals a deep connection between the new algorithm and classical iterative solvers, showing that the penetrating gradient generalizes Gauss‑Seidel to the nonlinear polynomial case.
The experimental section evaluates the combined Penetrating Gradient + Deepest Descent solver on a suite of 30 benchmark polynomial problems (including Rosenbrock, Himmelblau, Beale, and higher‑order synthetic systems) and on real‑world engineering applications such as inverse kinematics of robotic manipulators, power‑flow equations in electrical networks, and nonlinear reaction‑network equilibria. The solver is compared against Newton‑Raphson, Levenberg‑Marquardt, BFGS, and several global‑optimization heuristics. Results show a consistent reduction in the number of iterations (30 %–70 % fewer) and CPU time (20 %–50 % faster), while achieving final residuals below 10⁻⁸ in virtually all runs. Importantly, the method maintains a high success rate (>95 %) even when the initial guess is far from any solution, whereas traditional methods often diverge or stall in local minima for the same starts.
The authors acknowledge limitations: for systems with very high polynomial degree (≥10) the symbolic expansion and root‑finding become expensive, suggesting future work on adaptive degree reduction, sparse polynomial representations, or hybrid symbolic‑numeric schemes. They also propose learning‑based direction generators to enrich the candidate set in the Deepest Descent meta‑strategy, potentially further accelerating convergence.
In summary, the paper presents a compelling new paradigm for solving polynomial equation systems. By performing an exact, global‑optimal line search (the Penetrating Gradient) and by selecting the direction that yields the deepest descent, the method overcomes many of the pitfalls of conventional gradient‑based solvers, offers theoretical guarantees, and demonstrates superior empirical performance across a broad range of problems.