Characterization of regularity via variational stability of alternating projection sequences
本文证明了在希尔伯特空间中，两个闭凸集的正则性等价于交替投影方法在任意变分扰动下的收敛性，且无需最佳逼近集有界。
Abstract The notion of regular pair ( A , B ) for two nonempty closed convex subsets A and B of a Hilbert space $$\mathcal {H}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> was introduced by Borwein and Bauschke in 1993 to ensure convergence (in norm) of the alternating projection method to some point of the best approximation set. In 2022, De Bernardi and Miglierina showed that regularity of the pair ( A , B ) guarantees, additionally, the convergence for any variational perturbation of the alternating projection method, provided the corresponding best approximation sets are bounded. In this work, we show that the converse assertion is also true. Moreover, this converse assertion holds without requiring the best approximation sets to be bounded.