The Geometry of Simplex Degeneracy and Its Resolution via Perturbed Orbits

Simplex degeneracy is a persistent challenge in computational geometry and optimization, leading to algorithmic cycling and stalling. This paper explores the geometric underpinnings of degeneracy, characterizing it as the coincidence of more than the requisite number of bounding hyperplanes at a single vertex of a convex polytope. We introduce a novel conceptual framework for resolving such degeneracies: the method of Perturbed Orbits. This technique involves a symbolic infinitesimal perturbation of the simplex vertices, causing them to move along prescribed, non-intersecting curves or 'orbits.' This action effectively breaks the geometric coincidences, transforming a degenerate vertex into a cluster of non-degenerate vertices. The paper formalizes the geometry of degeneracy, details the mathematical construction of these perturbed orbits, and demonstrates that this method guarantees the resolution of ties and prevents cycling in algorithms like the simplex method. By providing a purely geometric interpretation, the perturbed orbits framework offers an intuitive and robust alternative to classical algebraic and lexicographical anti-degeneracy rules.