Summary: We analyze globally convergent derivative-free trust region algorithms relying on radial basis function interpolation models. Our results extend the recent work of A. R. Conn et al. [ibid. 20, No. 1, 387–415 (2009; Zbl 1187.65062)] to fully linear models that have a nonlinear term. We characterize the types of radial basis functions that fit in our analysis and thus show global convergence to first-order critical points for the ORBIT algorithm of the first author et al. [ibid. 30, No. 6, 3197–3219 (2008; Zbl 1178.65065)]. Using ORBIT, we present numerical results for different types of radial basis functions on a series of test problems. We also demonstrate the use of ORBIT in finding local minima on a computationally expensive environmental engineering problem.