A generalized complex power iteration-based algorithm for quartic–quadratic optimization problems over a spherical constraint and its applications

Quartic–quadratic optimization problems over the sphere have received a great attention in various real problems. Observations of three concrete applications primarily trigger our interest in this problem. This paper considers a generalized complex power method to solve the problem. Such a nonconvex optimization problem is studied by giving a suitable shift term to ensure convexity. Several convergence results are established, and we analyze the worst-case complexity in terms of the step-size and the number of iterations. We then apply our method to three tasks: (1) computing the ground state of Bose–Einstein condensation (BEC); (2) computing the quartic–quadratic optimization problems over the ellipsoid, which arises in the problem of computing the ground state of BEC by using a new discretization; and (3) approximating the desired transmit beampattern. The reported numerical results indicate that our method performs relatively quickly and effectively.