An analytic center quadratic cut method for the convex quadratic feasibility problem
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, N, of convex quadratic inequalities. We extend quadratic cut algorithm of Luo and Sun [3] for solving such problems by placing or translating the quadratic cuts directly through the current approximate center. We show that, in terms of total number of addition and translation of cuts, our algorithm has the same polynomial worst case complexity as theirs [3]. However, the total number of steps, where steps consist of (damped) Newton steps, function evaluations and arithmetic operations, required to update from one approximate center to another is O(1) + O(log(2) log(2) (N-2/epsilon(4))), where epsilon is the radius of the largest ball contained in the feasible set. 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 this case the quadratic cut algorithm is a fully polynomial approximation scheme. Furthermore, we show that, at each iteration, k, the total number steps (as described above) required to update from one approximate center to another is at most O(1) + O(log(2) log(2) (k(2)/epsilon(4))), with epsilon as defined above.
analytic center quadratic cuts convex quadratic feasibility problem interior-point methods potential function