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

TL;DR

This paper introduces a weakly polynomial-time algorithm for minimizing a non-convex quadratic function combined with logarithmic barriers within an $\, ext{l}_ ext{infty}$ trust-region, with potential applications in nonlinear programming.

Contribution

It presents a theoretical algorithm with weak polynomial complexity for a specific non-convex optimization problem involving logarithmic barriers.

Findings

Algorithm has weak polynomial time complexity.

Potential for practical acceleration discussed.

Application as step-directions in nonlinear programming.

Abstract

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.