Semidefinite relaxation approximation for multivariate bi‐quadratic optimization with quadratic constraints

Abstract

SUMMARYIn this paper, we consider the NP‐hard problem of finding global minimum of quadratically constrained multivariate bi‐quadratic optimization. We present some bounds of the considered problem via approximately solving the related bi‐linear semidefinite programming (SDP) relaxation. Based on the bi‐linear SDP relaxation, we also establish some approximation solution methods, which generalize the methods for the quadratic polynomial optimization in (SIAM J. Optim. 2003; 14:268–283). Finally, we present a special form, whose bi‐linear SDP relaxation can be approximately solved in polynomial time. Copyright © 2011 John Wiley & Sons, Ltd.