We consider the fundamental problem of maximizing a general quadratic function over an ellipsoidal domain, also known as the trust region problem. We give the first provable linear-time (in the number of non-zero entries of the input) algorithm for approximately solving this problem. Specifically, our algorithm returns an ǫ-approximate solution in timeÕ(N/ √ ǫ), where N is the number of non-zero entries in the input. This matches the runtime of Nesterov's accelerated gradient descent, suitable for the special case in which the quadratic function is concave, and the runtime of the Lanczos method which is applicable when the problem is purely quadratic.
One natural direction is to apply an approximate trust region solver, such as the linear-time solver of REF , to approximately solve the cubic regularization subroutine of Nesterov-Polyak.
A Linear-Time Algorithm for Trust Region Problems
ArXiv
ArXiv
Mathematics Computer Science