The Projected Polar Proximal Point Algorithm Converges Globally

TL;DR

This paper proves the global convergence of the projected polar proximal point algorithm (P4A) and its generalization (GP4A) using fixed point analysis and gauge representations, extending convergence guarantees in convex optimization.

Contribution

It introduces a generalized algorithm (GP4A) replacing the closed perspective transform with a closed gauge and provides convergence analysis and fixed point conditions for these methods.

Findings

GP4A and its under-relaxations exhibit global convergence when fixed points exist.

Convergence guarantees for P4A are derived as a special case of GP4A.

Fixed points correspond to global minimizers forming an exposed face of the fundamental set.

Abstract

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