A Counterexample to a Key Inequality and Structural Issues in a Modified Dai–Liao Conjugate-Gradient Analysis

This independent technical note examines the convergence analysis in A. Jaradat et al., “Modified Parameter of the Dai–Liao Conjugacy Condition of the Conjugate Gradient Method with Some Applications,” published in Nonlinear Dynamics and Systems Theory, 25(3) (2025), 266–287. We construct an explicit strongly convex quadratic counterexample satisfying the stated hypotheses of Lemma 3.1 and an initial strong Wolfe step, for which inequality (13) is false. Furthermore, an exact one-dimensional quadratic example with two algebraically self-consistent strong-Wolfe steps demonstrates that the proposed conjugate-gradient update parameter can be negative, contradicting the asserted nonnegativity property. We additionally identify an incorrect normalized-direction identity in Equation (20), a circular dependence involving the current step length in the proposed algorithm, and indexing inconsistencies and missing uniform estimates in the global-convergence arguments. These findings invalidate specific intermediate mathematical assertions and reveal substantial gaps in the published convergence analysis. They do not, however, establish divergence of the proposed optimization method. All counterexamples are supported by explicit mathematical calculations and an accompanying exact-arithmetic Python verification script.