  We investigate multiuser uplink communication from multiple single-antenna
users to a base station (BS), which is equipped with a movable-antenna (MA)
array and adopts zero-forcing receivers to decode multiple signals. We aim to
optimize the MAs' positions at the BS, to minimize the total transmit power of
all users subject to the minimum rate requirement. After applying
transformations, we show that the problem is equivalent to minimizing the sum
of each eigenvalue's reciprocal of a matrix, which is a function of all MAs'
positions. Subsequently, the projected gradient descent (PGD) method is
utilized to find a locally optimal solution. In particular, different from the
latest related work, we exploit the eigenvalue decomposition to successfully
derive a closed-form gradient for the PGD, which facilitates the practical
implementation greatly. We demonstrate by simulations that via careful
optimization for all MAs' positions in our proposed design, the total transmit
power of all users can be decreased significantly as compared to competitive
benchmarks.
