Abstract
The 2-sets convex feasibility problem aims at finding a point in the intersection of two closed convex sets A and B in a normed space X. More generally, we can consider the problem of finding (if possible) two points in A and B, respectively, which minimize the distance between the sets. In the present paper, we study some stability properties for the convex feasibility problem: we consider two sequences of sets, each of them converging, with respect to a suitable notion of set convergence, respectively, to A and B. Under appropriate assumptions on the original problem, we ensure that the solutions of the perturbed problems converge to a solution of the original problem. We consider both the finite-dimensional and the infinite-dimensional case. Moreover, we provide several examples that point out the role of our assumptions in the obtained results.
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References
Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer Monographs in Mathematics. Springer, New York (2003)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)
Bauschke, H.H., Combettes, P.L.: Convex analysis and monotone operator theory in Hilbert spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham (2017)
Borwein, J.M., Sims, B., Tam, M.K.: Norm convergence of realistic projection and reflection methods. Optimization 64, 161–178 (2015)
Borwein, J.M., Zhu, Q.J.: Techniques of variational analysis. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2005)
Censor, Y., Cegielski, A.: Projection methods: an annotated bibliography of books and reviews. Optimization 64, 2343–2358 (2015)
Combettes, P.L.: The Convex Feasibility Problem in Image Recovery, Vol. 95 of Advances in Imaging and Electron Physics. Academic Press, New York (1996)
Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach space theory. The basis for linear and nonlinear analysis. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2011)
Hundal, H.S.: An alternating projection that does not converge in norm. Nonlinear Anal. 57, 35–61 (2004)
Klee, V., Veselý, L., Zanco, C.: Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces. Stud. Math. 120, 191–204 (1996)
Lucchetti, R.: Convexity and well-posed problems. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2006)
Acknowledgements
The research of the first and second authors is partially supported by GNAMPA-INdAM. The research of the second and third authors is partially supported by Ministerio de Economía y Competitividad (Spain), MTM2015-68103-P, Plan Nacional de Matemáticas, (2016–2018). The authors thank the referees for useful remarks that helped them in preparing the final version of this paper.
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De Bernardi, C.A., Miglierina, E. & Molho, E. Stability of a convex feasibility problem. J Glob Optim 75, 1061–1077 (2019). https://doi.org/10.1007/s10898-019-00806-w
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DOI: https://doi.org/10.1007/s10898-019-00806-w