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Operator splitting techniques have recently gained popularity in convex optimization problems arising in various control fields. Being fixed-point iterations of nonexpansive (NE) operators, such metho...ds suffer many well known downsides, which include high sensitivity to ill conditioning and parameter selection, and consequent low accuracy and robustness. As universal solution we propose SuperMann, a Newton-type algorithm for finding fixed points of NE operators. It generalizes the classical Krasnosel'ski ˇı–Mann scheme, enjoys its favorable global convergence properties and requires exactly the same oracle. It is based on a novel separating hyperplane projection tailored for nonexpansive mappings which makes it possible to include steps along any direction. In particular, when the directions satisfy a Dennis–Mor´e condition we show that SuperMann converges superlinearly under mild assumptions, which, surprisingly, do not entail nonsingularity of the Jacobian at the solution butmerelymetric subregularity. As a result, SuperMann enhances and robustifies all operator splitting schemes for structured convex optimization, overcoming their well known sensitivity to ill conditioning.続きを見る
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