Application of a facial reduction algorithm and an inexact primal-dual path-following method for doubly nonnegative relaxation for mixed binary nonconvex quadratic optimization problems

In this paper, we propose an effective approach to solve doubly nonnegative relaxation (DNR) problems for mixed binary nonconvex quadratic optimization problems. In our approach, we convert a given DNR problem into another one that is smaller than the original one by exploiting degeneracy. We can expect numerical stability of primal-dual interior-point methods for the DNR problem to be improved because this conversion can be regarded as a partial application of a facial reduction algorithm. Moreover, we propose an inexact primal-dual path-following method for the resulting DNR problems. In our algorithm, to compute search directions, we solve large linear systems via the preconditioned symmetric quasi-minimal residual (PSQMR) method. To accelerate the convergence of the PSQMR method, we propose three preconditioners. Numerical results presented in this paper show that we can solve some instances of DNR problems quickly and accurately.