The projected polar proximal point algorithm converges globally

Friedlander, Macêdo, 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 gener...

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Bibliographic Details
Published in:Journal of global optimization 2022-09, Vol.84 (1), p.177-203
Main Author: Lindstrom, Scott B.
Format: Article
Language:English
Subjects:
Algorithms
Computer Science
Convergence
Euclidean space
Fixed points (mathematics)
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Real Functions
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Summary:Friedlander, Macêdo, 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.
ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-022-01136-0