Abstract
Maximally monotone operators and firmly nonexpansive mappings play key roles in modern optimization and nonlinear analysis. Five years ago, it was shown that if finitely many firmly nonexpansive operators are all asymptotically regular (i.e., they have or “almost have” fixed points), then the same is true for compositions and convex combinations. In this paper, we derive bounds on the magnitude of the minimal displacement vectors of compositions and of convex combinations in terms of the displacement vectors of the underlying operators. Our results completely generalize earlier works. Moreover, we present various examples illustrating that our bounds are sharp.
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Notes
We shall write \({\text {dom}}A = \big \{{x\in X}~\big |~{Ax\ne \varnothing }\big \}\) for the domain of A, \({\text {ran}}A = A(X) = \bigcup _{x\in X}Ax\) for the range of A, and \({\text {gra}}A=\big \{{(x,u)\in X\times X}~\big |~{u\in Ax}\big \}\) for the graph of A.
Here and elsewhere, \({\text {Id}}\) denotes the identity operator on X.
Given a nonempty closed convex subset C of X, we denote its projection mapping or projector by \(P_C\).
Let \(S:X\rightarrow X\). Then S is \(\alpha \)-averaged if there exists \(\alpha \in [0,1[\) such that \(S=(1-\alpha ){\text {Id}}+\alpha N\) and \(N:X\rightarrow X\) is nonexpansive.
We recall that a monotone operator \(B:X\rightrightarrows X\) is 3* monotone (see [9]) (this is also known as rectangular) if \((\forall (x,y^*)\in {\text {dom}}B\times {\text {ran}}B)\) \(\sup _{(z,z^*)\in {\text {gra}}B} \left\langle {x-z},{z^*-y^*} \right\rangle <+\infty \).
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Acknowledgements
The research of HHB was partially supported by a Discovery Grant of the Natural Sciences and Engineering Research Council of Canada. WMM was was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF Grant # CCF-1740425.
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Bauschke, H.H., Moursi, W.M. The magnitude of the minimal displacement vector for compositions and convex combinations of firmly nonexpansive mappings. Optim Lett 12, 1465–1474 (2018). https://doi.org/10.1007/s11590-018-1259-5
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DOI: https://doi.org/10.1007/s11590-018-1259-5