  When an iterative method is applied to solve the linear equation system in
interior point methods (IPMs), the attention is usually placed on accelerating
their convergence by designing appropriate preconditioners, but the linear
solver is applied as a black box with a standard termination criterion which
asks for a sufficient reduction of the residual in the linear system. Such an
approach often leads to an unnecessary 'oversolving' of linear equations. In
this paper, an IPM that relies on an inner termination criterion not based on
the residual of the linear system is introduced and analyzed. Moreover, new
indicators for the early termination of the inner iterations are derived from a
deep understanding of IPM needs. The new technique has been adapted to the
Conjugate Gradient (CG) and to the Minimum Residual method (MINRES) applied in
the IPM context. The new criterion has been tested on a set of quadratic
optimization problems including compressed sensing, image processing and
instances with partial differential equation constraints, and it has been
compared to standard residual tests with variable tolerance. Evidence gathered
from these computational experiments shows that the new technique delivers
significant improvements in terms of inner (linear) iterations and those
translate into significant savings of the IPM solution time.
