  Massive connectivity is a critical challenge of Internet of Things (IoT)
networks. In this paper, we consider the grant-free uplink transmission of an
IoT network with a multi-antenna base station (BS) and a large number of
single-antenna IoT devices. Due to the sporadic nature of IoT devices, we
formulate the joint activity detection and channel estimation (JADCE) problem
as a group-sparse matrix estimation problem. Although many algorithms have been
proposed to solve the JADCE problem, most of them are developed based on
compressive sensing technique, yielding suboptimal solutions. In this paper, we
first develop an efficient weighted $l_1$-norm minimization algorithm to better
approximate the group sparsity than the existing mixed $l_1/l_2$-norm
minimization. Although an enhanced estimation performance in terms of the mean
squared error (MSE) can be achieved, the weighted $l_1$-norm minimization
algorithm is still a convex relaxation of the original group-sparse matrix
estimation problem, yielding a suboptimal solution. To this end, we further
reformulate the JADCE problem as a mixed integer programming (MIP) problem,
which can be solved by using the branch-and-bound method. As a result, we are
able to obtain an optimal solution of the JADCE problem, which can be adopted
as an upper bound to evaluate the effectiveness of the existing algorithms.
Moreover, we also derive the minimum pilot sequence length required to fully
recover the estimated matrix in the noiseless scenario. Simulation results show
the performance gains of the proposed optimal algorithm over the proposed
weighted $l_1$-norm algorithm and the conventional mixed $l_1/l_2$-norm
algorithm. Results also show that the proposed algorithms require a short pilot
sequence than the conventional algorithm to achieve the same estimation
performance.
