Step-size restrictions and recentering bounds in the semidefinite linear complementarity problem: counterexamples and a corrected local analysis

This technical note examines the step-size estimates and recentering analysis in M. Achache and N. Boudiaf, “Complexity analysis of primal-dual algorithms for the semidefinite linear complementarity problem,” Rev. Anal. Numer. Theor. Approx. 40 (2011), no. 2, 95–106, DOI: 10.33993/jnaat402-1040. Exact scalar counterexamples show that the printed inequalities in Theorems 7 and 8 require additional restrictions, and that Lemma 10 cannot give a uniform recentering bound for arbitrary positive feasible input step sizes. A self-contained corrected analysis proves feasibility and the proximity estimate under 0 < alpha <= 1 and 2 alpha delta^2 < 1, verifies the adaptive rule alpha = 1/(4 delta^2) for delta >= 1, and supplies explicit integer bounds for both strict and non-strict stopping conventions. The note does not exclude polynomial complexity for an appropriately specified algorithm. This record contains the revised six-page PDF and a TeX source archive with a standard-library Python script for exact verification of the scalar counterexamples. ChatGPT assistance is disclosed in the manuscript.