An analytic center quadratic cut method for the convex quadratic feasibility problem (Q1396218)

The first problem studied in the article is to find a point in a convex set, which is contained in the unit ball and contains a full-dimensional ball of sufficiently small radius. The convex set is described by the intersection of a finite, yet large number of convex quadratic inequalities. The authors consider a quadratic cut method for solving the above quadratic feasible problem. At each iteration, the algorithm finds an approximate analytic center of an outer approximation of the convex set. If this approximate center is a solution, then the algorithm stops; otherwise, a quadratic inequality, violated at the current approximate center, is selected. If the violated inequality has already been added to the current outer approximation, then the corresponding quadratic cut is translated all the way to the current approximation center; otherwise, a new quadratic cut (corresponding to the violated inequality) is placed through the approximation center. Similar ideas have been applied to the second case, when the feasible set is an intersection of an infinite number of convex quadratic inequalities.