We introduce the Controlled Perturbation Algorithm (CPA) for escaping saddle points in generic non‑convex optimization problems. The computational cost of the algorithm is of order O(d), d is the number of degree of freedom. The key idea is to use two adaptive perturbations per coordinate, evaluate their directional derivatives, and deterministically select a descent direction – all without computing second or higher order derivatives. We also define the Non‑Descent Direction Approximation (NDDA) index as a cheap heuristic indicator of proximity to a local minimum.
This note is a preliminary algorithmic description intended to establish priority. No experimental validation is included here. A subsequent extended version will provide empirical results, code, and comparisons with existing methods. The algorithm is presented as a heuristic tool; rigorous convergence guarantees are left for future work.