Correction: A Bregman–Kaczmarz method for nonlinear systems of equations

Correction to: Computational Optimization and Applications https://doi.org/10.1007/s10589-023-00541-9

The original version of this article unfortunately contained a typo in Theorem 4.20.

In the first sentence, “and $M$-smooth” should be included at the end of the sentence so the correct sentence should read as below:

Let Assumption 1 hold true and let $\varphi$ be $\sigma$-strongly convex and $M$-smooth.

In item (ii) the first sentence, “Let $\varphi$ be additionally $M$-smooth and $\eta < \frac{\sigma }{2M}$.”

should read as follows

Let $\varphi$ be additionally $M$-smooth and let $\eta < \frac{\sigma }{2M}$.

In Eqn (33), the denominator on the right-hand side “$(1 + \eta )^{2} \kappa_{\min }^{2}$”

should be “$M(1 + \eta )^{2} \kappa_{\min }^{2}$”, so the correct equation should be

$$\frac{\sigma }{2}{\mathbb{E}}\left[ {\left\| {x_{k} - \hat{x}} \right\|_{2}^{2} } \right] \le {\mathbb{E}}\left[ {D_{\varphi } (x_{k} ,\hat{x})} \right] \le \left( {1 - \frac{{\sigma \left( {\frac{1}{2} - \eta } \right)p_{\min } }}{{M(1 + \eta )^{2} \kappa_{\min }^{2} }}} \right)^{k} {\mathbb{E}}\left[ {D_{\varphi } (x_{0} ,\hat{x})} \right].$$

(33)

The original article has been corrected.