  In this paper, we investigate a class of time-inconsistent discrete-time
stochastic linear-quadratic optimal control problems, whose time-consistent
solutions consist of an open-loop equilibrium control and a linear feedback
equilibrium strategy. The open-loop equilibrium control is defined for a given
initial pair, while the linear feedback equilibrium strategy is defined for all
the initial pairs. Maximum-principle-type necessary and sufficient conditions
containing stationary and convexity are derived for the existence of these two
time-consistent solutions, respectively. Furthermore, for the case where the
system matrices are independent of the initial time, we show that the existence
of the open-loop equilibrium control for a given initial pair is equivalent to
the solvability of a set of nonsymmetric generalized difference Riccati
equations and a set of linear difference equations. Moreover, the existence of
linear feedback equilibrium strategy is equivalent to the solvability of
another set of symmetric generalized difference Riccati equations.
