# Proof mining and the viscosity approximation method

## Laurentiu Leuştean

**Authors:** Paulo Firmino and Laurentiu Leuştean

This talk presents an application of proof mining to the viscosity approximation method (VAM) for accretive operators in Banach spaces, studied recently by Xu et al. [5]. Proof mining is a research program concerned with the extraction, by using proof-theoretic techniques, of new quantitative and qualitative information from mathematical proofs. We refer to Kohlenbach’s textbook [2] for details on proof mining.

Let \(X\) be a normed space and \(A:X\to 2^X\) be an accretive operator with a nonempty set of zeros. The VAMe iteration, a generalization of VAM obtained by adding error terms, is defined as follows:

\(\text{VAMe} \qquad x_0=x\in X, \quad x_{n+1}=\alpha_n f(x_n) +(1-\alpha_n)J_{\lambda_n}^Ax_n + e_n,\) where \(f:X\to X\) is an \(\alpha\)-contraction for \(\alpha\in[0,1)\), \((\alpha_n)_{n\in\mathbb{N}}\) is a sequence in \([0,1]\), \((\lambda_n)_{n\in\mathbb{N}}\) is a sequence in \((0,\infty)\), \((e_n)_{n\in\mathbb{N}}\) is a sequence in \(X\), and, for every \(n\in\mathbb{N}\), \(J_{\lambda_n}^A\) is the resolvent of order \(\lambda_n\) of \(A\).

In [1] we apply proof mining methods to obtain quantitative asymptotic regularity results for the VAMe iteration, providing uniform rates of asymptotic regularity, \(\left(J_{\lambda_n}^A\right)\)-asymptotic regularity and, for all \(m\in\mathbb{N}\), \(J_{\lambda_m}^A\)-asymptotic regularity of VAMe. For concrete instances of the parameter sequences, linear rates are computed by applying \cite[Lemma 2.8]{LeuPin23}, a slight variation of a lemma due to Sabach and Shtern [4].

## Bibliography

- P. Firmino, L. Leuştean,
*Quantitative asymptotic regularity of the VAM iteration with error terms for accretive operators in Banach spaces*, arXiv:2402.17947 [math.OC], 2024. - U. Kohlenbach,
, Springer Monographs in Mathematics, Springer, 2008.*Applied Proof Theory: Proof Interpretations and their Use in Mathematics* - L. Leuştean, P. Pinto
*Rates of asymptotic regularity for the alternating Halpern-Mann iteration*,, https://doi.org/10.1007/s11590-023-02002-y, 2023.*Optimization Letters* - S. Sabach, S. Shtern,
*A first order method for solving convex bilevel optimization problems*,,vol. 27 (2017), no. 2, pp. 640–660.*SIAM Journal on Optimization* - H.-K. Xu, N. Altwaijry, I. Alughaibi, S. Chebbi,
*The viscosity approximation method for accretive operators in Banach spaces*,, vol. 6 (2022), no. 1, pp. 37–50.*Journal of Nonlinear and Variational Analysis*