  This paper presents a universal trust-region method simultaneously
incorporating quadratic regularization and the ball constraint. We introduce a
novel mechanism to set the parameters in the proposed method that unifies the
analysis for convex and nonconvex optimization. Our method exhibits an
iteration complexity of $\tilde O(\epsilon^{-3/2})$ to find an approximate
second-order stationary point for nonconvex optimization. Meanwhile, the
analysis reveals that the universal method attains an $O(\epsilon^{-1/2})$
complexity bound for convex optimization and can be accelerated. These results
are complementary to the existing literature as the trust-region method was
historically conceived for nonconvex optimization. Finally, we develop an
adaptive universal method to address practical implementations. The numerical
results show the effectiveness of our method in both nonconvex and convex
problems.
