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.
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