The block coordinate descent (BCD) method is widely used for minimizing a continuous function f of several block variables. At each iteration of this method, a single block of variables is optimized, while the remaining variables are held fixed. To ensure the convergence of the BCD method, the subproblem to be optimized in each iteration needs to be solved exactly to its unique optimal solution. Unfortunately, these requirements are often too restrictive for many practical scenarios. In this paper, we study an alternative inexact BCD approach which updates the variable blocks by successively minimizing a sequence of approximations of f which are either locally tight upper bounds of f or strictly convex local approximations of f . We focus on characterizing the convergence properties for a fairly wide class of such methods, especially for the cases where the objective functions are either non-differentiable or nonconvex. Our results unify and extend the existing convergence results for many classical algorithms such as the BCD method, the difference of convex functions (DC) method, the expectation maximization (EM) algorithm, as well as the alternating proximal minimization algorithm.

Abstract. The alternating direction method of multipliers (ADMM) is widely used to solve large-scale linearly constrained optimization problems, convex or nonconvex, in many engineering fields. However there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex. In this paper we analyze the convergence of the ADMM for solving certain nonconvex consensus and sharing problems. We show that the classical ADMM converges to the set of stationary solutions, provided that the penalty parameter in the augmented Lagrangian is chosen to be sufficiently large. For the sharing problems, we show that the ADMM is convergent regardless of the number of variable blocks. Our analysis does not impose any assumptions on the iterates generated by the algorithm, and is broadly applicable to many ADMM variants involving proximal update rules and various flexible block selection rules.

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We consider the problem of linear transceiver design to achieve max-min fairness in a downlink MIMO multicell network. This problem can be formulated as maximizing the minimum rate among all the users in an interfering broadcast channel (IBC). In this paper we show that when the number of antennas is at least two at each of the transmitters and the receivers, the min rate maximization problem is NP-hard in the number of users. Moreover, we develop a low-complexity algorithm for this problem by iteratively solving a sequence of convex subproblems, and establish its global convergence to a stationary point of the original minimum rate maximization problem. Numerical simulations show that this algorithm is efficient in achieving fairness among all the users. I. INTRODUCTIONWe consider the linear transceiver design problem in a MIMO-IBC, in which a set of Base Stations (BSs) send data to their intended users. Both the BSs and the users are equipped with multiple antennas, and they share the same time/frequency resource for transmission. The objective is to maximize the minimum rate among all the users in the network, in order to achieve network-wide fairness.Providing max-min fairness has long been considered as an important design criterion for wireless networks. Hence various algorithms that optimize the min-rate utility in different network settings have been proposed in the literature. References [20], [21] are early works that studied the max-min signal to interference plus noise ratio (SINR) power control problem and a related SINR feasibility problem in a scalar interference channel (IC). It was shown in [20], [21] that for randomly generated scalar ICs, with probability one there exists a unique optimal solution to the max-min problem. The proposed algorithm with an additional binary search can be used to solve the max-min fairness problem efficiently. Recently reference [17] derived a set of algorithms based on nonlinear Perron-Frobenius theory for the same network setting. Differently from [20], [21], the proposed algorithms can also deal with individual users' power constraints. Apart from the scalar IC case, there have been many published results [1], [3], [4], [8], [14], [18], [19] on the min rate maximization problem in a multiple input single output (MISO) network, in which the BSs are equipped with multiple antennas and the users are only equipped with a single antenna. Reference [19] utilized the nonnegnative matrix theory to study the related power control problem when the beamformers are known and fixed. When optimizing the transmit power and the beamformers jointly, M. Razaviyayn, M. Hong, and Z.-Q. Luo are with the