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The problem of finding a point in the intersection of closed sets can besolved by the method of alternating projections and its variants. It was shownin earlier papers that for convex sets, the strategy of using quadraticprogramming (QP) to project onto the intersection of supporting halfspacesgenerated earlier by the projection process can lead to an algorithm thatconverges multiple-term superlinearly. The main contributions of this paper areto show that this strategy can be effective for super-regular sets, which arestructured nonconvex sets introduced by Lewis, Luke and Malick. Manifoldsshould be approximated by hyperplanes rather than halfspaces. We prove thelinear convergence of this strategy, followed by proving that superlinear andquadratic convergence can be obtained when the problem is similar to thesetting of the Newton method. We also show an algorithm that converges at anarbitrarily fast linear rate if halfspaces from older iterations are used toconstruct the QP.

Nonconvex set intersection problems: From projection methods to the Newton method for super-regular sets