New global algorithms for quadratic programming with a few negative eigenvalues based on alternative direction method and convex relaxation - Luo, Hezhi; Bai, Xiaodi; Lim, Gino; Peng, Jiming (Mathematical programming computation, 2019)
We consider a quadratic program with a few negative eigenvalues (QP- r -NE) subject to linear and convex quadratic constraints that covers many applications and is known to be NP-hard even with one negative eigenvalue (QP1NE). In this paper, we first introduce a new global algorithm (ADMBB), which integrates several simple optimization techniques such as alternative direction method, and branch-and-bound, to find a globally optimal solution to the underlying QP within a pre-specified ϵ -tolerance. We establish the convergence of the ADMBB algorithm and estimate its complexity. Second, we develop a global search algorithm (GSA) for QP1NE that can locate an optimal solution to QP1NE within ϵ -tolerance and estimate the worst-case complexity bound of the GSA. Preliminary numerical results demonstrate that the ADMBB algorithm can effectively find a global optimal solution to large-scale QP- r -NE instances when r ≤ 10 , and the GSA outperforms the ADMBB for most of the tested QP1NE instances. The software reviewed as part of this submission was given the DOI (digital object identifier) https://doi.org/10.5281/zenodo.1344739 .