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The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets$$\\{A_n\\}$$<\/jats:tex-math>{<\/mml:mo>A<\/mml:mi>n<\/mml:mi><\/mml:msub>}<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and$$\\{B_n\\}$$<\/jats:tex-math>{<\/mml:mo>B<\/mml:mi>n<\/mml:mi><\/mml:msub>}<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, each of them converging, with respect to the Attouch-Wets variational convergence, respectively, toA<\/jats:italic>andB<\/jats:italic>. Given a starting point$$a_0$$<\/jats:tex-math>a<\/mml:mi>0<\/mml:mn><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we consider the sequences of points obtained by projecting on the \u201cperturbed\u201d sets, i.e., the sequences$$\\{a_n\\}$$<\/jats:tex-math>{<\/mml:mo>a<\/mml:mi>n<\/mml:mi><\/mml:msub>}<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and$$\\{b_n\\}$$<\/jats:tex-math>{<\/mml:mo>b<\/mml:mi>n<\/mml:mi><\/mml:msub>}<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>given by$$b_n=P_{B_n}(a_{n-1})$$<\/jats:tex-math>b<\/mml:mi>n<\/mml:mi><\/mml:msub>=<\/mml:mo>P<\/mml:mi>B<\/mml:mi>n<\/mml:mi><\/mml:msub><\/mml:msub>(<\/mml:mo>a<\/mml:mi>n<\/mml:mi>-<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msub>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and$$a_n=P_{A_n}(b_n)$$<\/jats:tex-math>a<\/mml:mi>n<\/mml:mi><\/mml:msub>=<\/mml:mo>P<\/mml:mi>A<\/mml:mi>n<\/mml:mi><\/mml:msub><\/mml:msub>(<\/mml:mo>b<\/mml:mi>n<\/mml:mi><\/mml:msub>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences$$\\{a_n\\}$$<\/jats:tex-math>{<\/mml:mo>a<\/mml:mi>n<\/mml:mi><\/mml:msub>}<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and$$\\{b_n\\}$$<\/jats:tex-math>{<\/mml:mo>b<\/mml:mi>n<\/mml:mi><\/mml:msub>}<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>converge in norm to a point in the intersection ofA<\/jats:italic>andB<\/jats:italic>. In particular, we consider both when the intersection$$A\\cap B$$<\/jats:tex-math>A<\/mml:mi>\u2229<\/mml:mo>B<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>reduces to a singleton and when the interior of$$A \\cap B$$<\/jats:tex-math>A<\/mml:mi>\u2229<\/mml:mo>B<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is nonempty. Finally we consider the case in which the limit setsA<\/jats:italic>andB<\/jats:italic>are subspaces.<\/jats:p>","DOI":"10.1007\/s10898-021-01025-y","type":"journal-article","created":{"date-parts":[[2021,4,23]],"date-time":"2021-04-23T04:02:56Z","timestamp":1619150576000},"page":"323-350","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A variational approach to the alternating projections method"],"prefix":"10.1007","volume":"81","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9654-1324","authenticated-orcid":false,"given":"Carlo Alberto","family":"De Bernardi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Enrico","family":"Miglierina","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,4,23]]},"reference":[{"key":"1025_CR1","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1137\/S0036144593251710","volume":"38","author":"HH Bauschke","year":"1996","unstructured":"Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. 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