A note on approximating quadratic programming with rank constraint

Abstract

The quadratic maximization problem with rank constraint means where n and m are two given positive integers with n ≤ m, Q is a given real symmetric matrix, is the unit sphere in and stands for the Euclidean inner product. In this note, motivated by Briët, Filho and Vallentin's and Ye's excellent works [J. Briët, F. Filho, and F. Vallentin, The positive semidefinite Grothendieck problem with rank constraint, Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6198, 2010, pp. 31–42.; Y. Ye, Approximating quadratic programming with bound and quadratic constraints, Math. Program. 84 (1999), pp. 219–226], we first derive the relative approximation ratio for the above problem with arbitrary symmetric matrix Q. While Q was supposed to be positive semidefinite in [J. Briët, F. Filhom, and F. Vallentin, The positive semidefinite Grothendieck problem with rank constraint, Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 6198, 2010, pp. 31–42.]. Secondly, we extend such a relative approximation ratio to a generalization of the above optimization problem. Finally, a slightly improved relative approximation ratio for the optimization problem considered in [Y. Ye, Approximating quadratic programming with bound and quadratic constraints, Math. Program. 84 (1999), pp. 219–226] is presented.