A finite-step breakdown of a Dai–Liao-type conjugate gradient algorithm on a uniformly convex polynomial

This technical preprint examines Algorithm 2.1 in Y. Zheng, RAIRO–Operations Research 55 (2021), 3281–3291, DOI 10.1051/ro/2021159. A uniformly convex two-dimensional quartic objective and five rational step lengths give an admissible finite trajectory that generates an ascent direction at a nonstationary point. The next strong Wolfe curvature test is then impossible for every positive step. Exact rational certificates are provided for t = 1 and the interior value t = 0.99999. Both the literally printed decrease inequality and the standard Armijo inequality are checked. The result identifies a failure of well-definedness of the algorithm as stated. It does not refute Theorem 3.5 under its explicit sufficient-descent hypothesis, nor a convergence statement conditioned on an existing infinite run. A direction safeguard and a self-contained sufficient convergence proof are included. The files contain the note, LaTeX source, Python standard-library verifier, exact-check outputs, and reproduction instructions. AI disclosure: OpenAI ChatGPT was used for English-language improvements, numerical and exact-arithmetic checks, and assistance with mathematical exploration and drafting. The author takes full responsibility for the counterexamples, theorem and proposition statements, proofs, computations, interpretations, and all other content. The AI system is not an author; its output is not a substitute for mathematical proof or the executable exact certificate.