An indefinite stochastic linear-quadratic(LQ) optimal control problem with cross term over an infinite time horizon is studied, allowing the weighting matrices to be indefinite. A systematic approach to the problem based on semidefinite programming (SDP) and .elated duality analysis is developed. Several implication relations among the SDP complementary duality, the existence of the solution to the generalized Riccati equation and the optimality of LQ problem are discussed. Based on these relations, a numerical procedure that provides a thorough treatment of the LQ problem via primal-dual SDP is given: it identifies a stabilizing optimal feedback control or determines the problem has no optimal solution. An example is provided to illustrate the results obtained.