Strong Convergence of Alternating Projections

• Published in: Journal of optimization theory and applications Vol. 194; no. 1; pp. 306 - 324
• Main Authors: Melo, Ítalo Dowell Lira, da Cruz Neto, João Xavier, de Brito, José Márcio Machado
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Convergence; Convexity; Engineering; Hilbert space; Manifolds; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Theorems; Theory of Computation; Alternating projections; Strong convergence; Convex feasibility problem; Bernoulli measure; Quasi-normal sequence; 28D05; 47H09; Hadamard space; 46N10
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-022-02028-9

In this paper, we provide a necessary and sufficient condition under which the method of alternating projections on Hadamard spaces converges strongly. This result is new even in the context of Hilbert spaces. In particular, we found the circumstance under which the iteration of a point by projections converges strongly and we answer partially the main question that motivated Bruck’s paper (J Math Anal Appl 88:319–322, 1982). We apply this condition to generalize Prager’s theorem for Hadamard manifolds and generalize Sakai’s theorem for a larger class of the sequences with full measure with respect to Bernoulli measure. In particular, we answer to a long-standing open problem concerning the convergence of the successive projection method (Aleyner and Reich in J Convex Anal 16:633–640, 2009). Furthermore, we study the method of alternating projections for a nested decreasing sequence of convex sets on Hadamard manifolds, and we obtain an alternative proof of the convergence of the proximal point method.