On the convergence of the method of analytic centers when applied to convex quadratic programs

AbstractThis work examines the method of analytic centers of Sonnevend when applied to solve generalized convex quadratic programs — where also the constraints are given by convex quadratic functions. We establish the existence of a two-sided ellipsoidal approximation for the set of feasible points around its center and show, that a simple (zero order) algorithm starting from an initial center of the feasible set generates a sequence of strictly feasible points whose objective function values converge to the optimal value. Concerning the speed of convergence it is shown that an upper bound for the gap in between the objective function value and the optimal value is reduced by a factor ofε with $$O(\sqrt m \left| {ln \varepsilon } \right|)$$ iterations wherem is the number of inequality constraints. Here, each iteration involves the computation of one Newton step. The bound of $$O(\sqrt m \left| {ln \varepsilon } \right|)$$ Newton iterations to guarantee an error reduction by a factor ofε in the objective function is as good as the one currently given forlinear programs. However, the algorithm considered here is of theoretical interest only, full efficiency of the method can only be obtained when accelerating it by some (higher order) extrapolation scheme, see e.g. the work of Jarre, Sonnevend and Stoer.