On Four Issues in the Complexity Analysis of a Parametric Kernel Interior-Point Method for Convex Quadratic Programming

We re-examine the complexity analysis of R. Chalekh and E. A. Djeffal, “Complexity Analysis of an Interior-point Algorithm for CQP Based on a New Parametric Kernel Function,” Statistics, Optimization & Information Computing 12 (2024), 153–166. Four distinct issues are identified. First, the first derivative displayed in equation (12) is not the derivative of the kernel defined in equation (1), and the discrepancy propagates to the displayed second and third derivatives. Second, the assertion \(f(t)>0\) in Lemma 1 is false for every admissible \(p≥ 4\) when \(t\) is sufficiently large. Third, after differentiating the kernel correctly, the claimed global strict convexity itself fails for sufficiently large \(p\); in particular, \(ψ_h''(1.02)<0\) for \(p=100\). This contradicts the paper’s stated kernel-function requirements and is relevant because the large-update choice \(p=(log n)/2-1\) is unbounded as \(n→∞\). Fourth, the displayed algorithm computes search directions from the logarithmic-barrier Newton system, whereas equation (42) uses the descent identity associated with a new-kernel search direction and a different proximity measure. The same identification invalidates the proof, and in fact the statement, of Lemma 5. Consequently, the published proof does not establish the stated complexity bound for the displayed algorithm. We do not claim that a suitably modified method cannot attain that bound; a repair would require a revised search system and a new analysis based on the correct derivatives.