  Trust-region methods (TR) can converge quadratically to minima where the
Hessian is positive definite. However, if the minima are not isolated, then the
Hessian there cannot be positive definite. The weaker
Polyak$\unicode{x2013}${\L}ojasiewicz (P{\L}) condition is compatible with
non-isolated minima, and it is enough for many algorithms to preserve good
local behavior. Yet, TR with an $\textit{exact}$ subproblem solver lacks even
basic features such as a capture theorem under P{\L}.
  In practice, a popular $\textit{inexact}$ subproblem solver is the truncated
conjugate gradient method (tCG). Empirically, TR-tCG exhibits super-linear
convergence under P{\L}. We confirm this theoretically.
  The main mathematical obstacle is that, under P{\L}, at points arbitrarily
close to minima, the Hessian has vanishingly small, possibly negative
eigenvalues. Thus, tCG is applied to ill-conditioned, indefinite systems. Yet,
the core theory underlying tCG is that of CG, which assumes a positive definite
operator. Accordingly, we develop new tools to analyze the dynamics of CG in
the presence of small eigenvalues of any sign, for the regime of interest to
TR-tCG.
