Over-Relaxed Projected-Forward Iterations for Cocoercive Variational Inequalities: Active-Face Spectral Tuning

<title>Abstract</title> <p>Westudy outer relaxation of the projected-forward iteration for cocoercive variational inequalities. The admissible over-relaxation interval follows from standard forward–backward averagedness, so the issue addressed here is parameter selection rather than global admissibility. For box constraints, strict complementarity and local differentiability imply finite identification of the active face. After identification, the relaxed map has a block triangular derivative; when the free–free Jacobian is symmetric positive definite, its local spectral radius is the maximum of the active factor |1−𝜆| and the free-face factors |1−𝜆𝛾𝜃𝑗|. This yields the closed-form spectral-radius minimizer 𝜆∗ loc = 2/(𝑎𝛾 +𝑏𝛾) and the exact criterion 𝑎𝛾 + 𝑏𝛾 &lt; 2 for genuine over-relaxation. Joint tuning shows that an unrelaxed member is locally optimal when the forward step can be freely retuned, which identifies fixed or globally constrained forward steps as the natural regime for outer relaxation. We propose a pattern-reset safeguarded selector that clips spectral estimates to the global averagedness interval. In a dense 120-dimensional benchmark family where high-curvature directions are active, the locally preferred forward step violates the global step bound, while the adaptive relaxation reduces iteration counts by about 33% across three conditioning levels. A nonlinear 80-dimensional test confirms the same active-face mechanism and also shows the advantage of retuning the forward step when that option is globally admissible. 2020 Mathematics Subject Classification: 47H05, 47J20, 47J25, 49J40, 65K15, 90C25.</p>