We extend traditional complexity analyses of trust-region methods for unconstrained, possibly nonconvex, optimization. Whereas most complexity analyses assume uniform boundedness of the model Hessians, we work with potentially unbounded model Hessians. Boundedness is not guaranteed in practical implementations, in particular ones based on quasi-Newton updates such as PSB, BFGS and SR1. Our analysis is conducted for a family of trust-region methods that includes most known methods as special cases. We examine two regimes of Hessian growth: one bounded by a power of the number of successful iterations, and one bounded by a power of the number of iterations. This allows us to formalize and address the intuition of Powell (Math. Program 29:297–303, 1984), who studied convergence under a special case of our assumptions, but whose proof contained complexity arguments. Specifically, for \(0 \le p < 1\) , we establish sharp \(O([(1-p)\epsilon ^{-2}]^{1/(1-p)})\) evaluation complexity to find an \(\epsilon \) -stationary point when model Hessians are \(O(|\mathcal {S}_{k-1}|^p)\) , where \(|\mathcal {S}_{k-1}|\) is the number of iterations where the step was accepted, up to iteration \(k-1\) . For \(p = 1\) , which is the case studied by Powell, we establish a sharp \(O(\exp (c_1\epsilon ^{-2}))\) evaluation complexity for a certain constant \(c_1 > 0\) . This is far better than the double exponential bound that Powell (Math. Program 29:297–303, 1984) suspected, and is far worse than other bounds surmised elsewhere in the literature. We establish similar sharp bounds when model Hessians are \(O(k^p)\) , where \(k\) is the iteration counter, for \(0 \le p < 1\) . When \(p = 1\) , the complexity bound depends on the parameters of the family, but reduces to \(O((1 - \log (\epsilon ))\exp (c_2\epsilon ^{-2}))\) for a certain constant \(c_2 > 0\) for the special case of the standard trust-region method. As special cases, we derive novel complexity bounds for (strongly) convex objectives under the same growth assumptions.
