A method of Analytic Centers for Quadratically Constrained Convex Quadratic Programs

Abstract

An interior point method is developed for maximizing a concave quadratic function under convex quadratic constraints. The algorithm constructs a sequence of nested convex sets and finds their approximate centers using a partial Newton step. Given the first convex set and its approximate center, the total arithmetic operations required to converge to an approximate solution are of order $O(\sqrt m (m + n)n^2 \ln \varepsilon )$, where m is the number of constraints, n is the number of variables, and $\varepsilon $ is determined by the desired tolerance of the optimal value and the size of the first convex set. A method to initialize the algorithm is also proposed so that the algorithm can start from an arbitrary (perhaps infeasible) point.