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Register a new userThe Projected Polar Proximal Point Algorithm Converges Globally
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Scott B. Lindstrom
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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.
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We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity.
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