Weakly polynomial efficient minimization of a non-convex quadratic function with logarithmic barriers in a trust-region

We introduce a particular optimization problem that minimizes the sum of a non-convex quadratic function and logarithmic barrier-functions in a ${\mathrm{\ell }}_{\mathrm{\infty }}$-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.