An Analytic Center Quadratic Cut Method for the Convex Quadratic Feasibility Problem.

Faranak Sharifi Mokhtarian and Jean-Louis Goffin

We consider a quadratic cut method based on analytic centers for two cases of convex quadratic feasibility problems. In the first case, the convex set is defined by a finite yet large number of convex quadratic inequalities. We extend quadratic cut algorithm of Luo and Sun~\cite{LS1} for solving such problems by placing or translating the quadratic cuts directly through the current approximate center. We show that our algorithm has the same polynomial worst case complexity as theirs~\cite{LS1}. In the second case, the convex set is defined by an infinite number of certain strongly convex quadratic inequalities. We adapt the same quadratic cut method for the first case to the second one. We show that in the second case the quadratic cut algorithm is a fully polynomial approximation scheme. Furthermore, we show that in both cases, at each iteration, the total number of (damped) Newton steps required to update from one approximate analytic center to another is at most $O(1)$.

GERAD Tech. report G-2000-18, 19 pp, april 2000.