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Friedlander, Mac\^{e}do, and Pong recently introduced the projected polarproximal point algorithm (P4A) for solving optimization problems by using theclosed perspective transforms of convex objectives. We analyse a generalization(GP4A) which replaces the closed perspective transform with a more generalclosed gauge. We decompose GP4A into the iterative application of two separateoperators, and analyse it as a splitting method. By showing that GP4A and itsunder-relaxations exhibit global convergence whenever a fixed point exists, weobtain convergence guarantees for P4A by letting the gauge specify to theclosed perspective transform for a convex function. We then provideeasy-to-verify sufficient conditions for the existence of fixed points for theGP4A, using the Minkowski function representation of the gauge. Conveniently,the approach reveals that global minimizers of the objective function for P4Aform an exposed face of the dilated fundamental set of the closed perspectivetransform.

The Projected Polar Proximal Point Algorithm Converges Globally