enough in the respective parameter region. Hence, we require an approach which goes beyond the classical offline/online decomposition. Recently, RB methods have been advised with a progressive construction of ROMs [@BMV2020; @GHH2016; @ZF2015]. Also localized RB methods that are based on efficient localized a posteriori error control and online enrichment [@BEOR2017; @OS2015] overcome traditional offline/online splitting and are thus particularly well suited for applications in optimization or inverse problems [@OSS2018; @OS2017].

**Trust-Region reduced order models for second-order methods.** Trust-Region (TR) approaches are a class of optimization methods that are advantageous for the usage of locally accurate surrogate models. The main idea is to solve optimization sub-problems only in a local area of the parameter set which resolves the burden of constructing a global RB space. The problem that might occur is the fact that during this minimization one usually moves away from the original parameters on which the reduced order model was built, and the quality of the reduced model cannot be guaranteed anymore. For that reason, a priori and a posteriori error analysis are required to ensure accurate reduced order approximations for the optimization problem; cf. [@GV17; @HV08; @KTV13]. In [@AFS00; @SS13] a TR approach was proposed to control 