O CEMS.UL - Centro de Estudos Matemáticos promove a realização do seminário "Quantitative results on a generalized viscosity approximation method", com a participação de Paulo Firmino (Ciências ULisboa e CEMS.UL).
Abstract: In this talk we present the results of a quantitative analysis of the asymptotic behaviour of the generalized Viscosity Approximation Method (genVAM), an iteration we studied in joint work with Leustean. The genVAM iteration is a generalization of VAM, an iteration studied in Banach spaces by Xu et al. [6], to W-hyperbolic spaces and to families (T_n) of mappings satisfying certain resolvent-like conditions (introduced by Leustean, Nicolae and Sipos [5] in their abstract analysis of the Proximal Point Algorithm). The genVAM iteration also generalizes the abstract HPPA studied by Aoyama, Kimura, Takahashi, and Toyoda [1], itself a generalization of the well-known Halpern-type Proximal Point Algorithm. We computed rates of ((T_n)-)asymptotic regularity, as well as rates of T_m-asymptotic regularity for all m. In this way, we extend to this very general setting the quantitative asymptotic regularity results we had obtained for the VAM iteration in Banach spaces [2]. Furthermore, as an application of the lemma due to Sabach and Shtern, linear such rates are computed for particular choices of the parameter sequences. On the setting of complete CAT(0) spaces, we compute rates of metastability for the abstract HPPA. We then, relying on results by Kohlenbach and Pinto [4], compute rates of metastability for genVAM, in doing so establishing a qualitative convergence result for the iteration. This talk reports on [3], recent joint work with Leustean.
References:
[1] K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda. Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Analysis, 67:2350-2360, 2007.
[2] P. Firmino and L. Leustean. Quantitative asymptotic regularity of the VAM iteration with error terms for m-accretive operators in Banach spaces. Zeitschrift für Analysis und ihre Anwendugen, 44:501-519, 2025.
[3] P. Firmino, L. Leustean. Quantitative results on a generalized viscosity approximation method. arXiv:2512.09968 [math.OC], 2025.
[4] U. Kohlenbach and P. Pinto. Quantitative translations for viscosity approximation methods in hyperbolic spaces. Journal of Mathematical Analysis and Applications, 507:125823, 2022.
[5] L. Leustean, A. Nicolae, and A. Sipos. An abstract proximal point algorithm. Journal of Global Optimization, 72:553-577, 2018.
[6] H.-K. Xu, N. Altwaijry, I. Alughaibi, and S. Chebbi. The viscosity approximation method for accretive operators in Banach spaces. Journal of Nonlinear and Variational Analysis, 6(1):37–50, 2022.