  This paper first presents necessary and sufficient conditions for the
solvability of discrete time, mean-field, stochastic linear-quadratic optimal
control problems. Then, by introducing several sequences of bounded linear
operators, the problem becomes an operator stochastic LQ problem, in which the
optimal control is a linear state feedback. Furthermore, from the form of the
optimal control, the problem changes to a matrix dynamic optimization problem.
Solving this optimization problem, we obtain the optimal feedback gain and thus
the optimal control. Finally, by completing the square, the optimality of the
above control is validated.
