The Projected Polar Proximal Point Algorithm Converges Globally

The paper establishes global convergence of GP4A and its under-relaxed variants to a fixed point whenever the fixed-point set is nonempty (Theorem 4.11 for γ=0; Corollary 4.12 for 0≤γ<1), and it supplies concrete, easily checked conditions ensuring existence of fixed points (Theorem 5.2). Specializing to κ=f^π recovers that λ converges to λ′/(1+λ′) with λ′=1/inf f and identifies the scaled minimizers (Theorem 5.3), as well as the convergence of shadow components λ_n (Corollary 4.13) . By contrast, the candidate solution incorrectly relies on an “averagedness” claim for N=P_S∘T based only on Lipschitz continuity, and it asserts that Picard convergence (γ=0) is unavailable in general—statements contradicted by the paper’s strict QNE analysis and convergence theorems. The paper also shows PS∘T is not generically a cutter (hence not necessarily averaged), invalidating the model’s core argument .