  We introduce a particular optimization problem that minimizes the sum of a
non-convex quadratic function and logarithmic barrier-functions in a
$\ell_\infty$-trust-region (i.e. cube). Our paper covers three topics.
  We explain the relevance of the considered problem. We lay out how solutions
of this problem can be used as efficient step-directions in solution methods
for nonlinear programming.
  We present a theoretical algorithm for solving the problem. We show that this
algorithm has weak polynomial time-complexity.
  A practical method is under development. In the outlook we discuss how the
given method can be accelerated for better practical performance. We also lay
out where the difficulties live when trying to formulate an accelerated
primal-dual variant.
