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Modulus of regularity and rate of convergence for a Fejer monotone sequence Lopez, Genaro


Many problems in applied mathematics can be brought into the following format:

Let $(X,d)$ be a metric space and $F:X\to \mathbb{R}$ be a function: find a zero $z\in X$ of $F.$

This statement covers many equilibrium, fixed points and minimization problems. Numerical methods, e.g. based on suitable iterative techniques, usually yield sequences $(x_n)$ in $X$ of approximate zeros, i.e. $|F(x_n)|< 1/n.$ Based on extra assumptions (e.g. the compactness of $X,$ the Fejér monotonicity of $(x_n)$ and the continuity of $F$) one then shows that $(x_n)$ converges to an actual zero $z$ of $F.$ An obvious question then concerns the speed of the convergence of $(x_n)$ towards $z$ and whether there is an effective rate of convergence.
Even though sometimes left implicit, the effectivity of iterative procedures in the case of unique zeros rests on the existence of an effective so-called modulus of uniqueness, see [Kohlenbach]. In this talk, we are concerned with a generalization of the concept of "modulus of uniqueness", called "modulus of regularity" which is applicable also in the non-unique case. While the concept of a modulus of regularity has been used in various special situations before (see e.g. [Anderson] and the literature cited there), we develop it here as a general tool towards a unified treatment of a number of concepts studied in convex optimization such as metric subregularity, H\"older regularity, weak sharp minima etc. which can be seen as instances of the concept of regularity w.r.t. $\text{zer}\;F$ for suitable choices of $F.$
This talk is based on a still in progress joint work with A. Nicolae and U. Kohlenbach.

[Anderson] R.M. Anderson, "Almost" implies "Near", Trans. Amer. Math. Soc. 296 (1986), 229-237.
[Kohlenbach] U. Kohlenbach, Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation, Ann. Pure Appl. Logic 64 (1993), 27-94.

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