The broadcast nature of wireless transmissions makes wireless networks susceptible to interference, which deteriorates quality of service (QoS) provisioning. Power control is considered as a promising technique to mitigate interference. One primary objective of power control is to maximize the system utility that can achieve a variety of fairness objectives among users[1–4]. However, maximizing the system utility, under the physical interference model, often involves nonconvex optimization and it is known to be NP-hard, due to the complicated coupling among users through mutual interference effects[5].

Due to the nonconvex nature of the power control problem, it is challenging to find the globally optimal power allocation in a distributed manner. Notably, the authors of[6–9] devised distributed power control algorithms to find power allocations that can only satisfy the local optimality conditions, but global optimality could not be guaranteed in general, except for some special convexifiable cases (e.g., with strictly increasing log-concave utility functions). Another thread of work applied game-theoretic approaches to power control by treating it as a noncooperative game among transmitters[10, 11]. However, distributed solutions that converge to a Nash equilibrium may be suboptimal in terms of maximizing the total system utility. Different from these approaches, the authors of[12] transformed the power control problem into a DC (difference of convex functions) optimization problem[13]. Then, the global optimal solution can be solved in a centralized manner with the branch-and-bound algorithm. Recent study[14] proposed a globally optimal power control scheme, named MAPEL, by exploiting the monotonic nature of the underlying optimization problem. However, the complexity and the centralized nature of MAPEL hinder its applicability in practical scenarios, and thus it can be treated rather as a benchmark for performance evaluation in distributed networks.

To find the globally optimal power allocation in a distributed setting, recent study[15] has proposed the SEER algorithm based on Gibbs sampling[16], which can approach the globally optimal solution in an asymptotic sense when the control parameter in Gibbs sampling tends to infinity. Notably, for each iteration in the SEER algorithm, each user utilizes Gibbs sampling to compute its transition probability distribution for updating its transmission power, where the requirement for message passing and computing the transition probability distribution in each iteration can be demanding when applied to ad hoc communications without centralized control.

A challenging task in distributed power control in ad hoc networks is to reduce the amount of message passing while preserving the global optimality. To tackle this challenge, we first show that the globally optimal point lies on the boundary of the feasible region. This property is utilized to transform the utility maximization problem into an equivalent max–min problem with more structure, which can be solved by combining recent advances in extended duality theory (EDT)[17] with simulated annealing (SA)[18]. Compared with the classical duality theory with nonzero duality gap for nonconvex optimization problems, EDT can guarantee zero duality gap between the primal and dual problems by utilizing nonlinear Lagrangian functions. This property allows for solving the nonconvex problem by its *extended dual* while preserving the global optimality with distributed implementation. Furthermore, as will be shown in Section “Power control for unicast communications”, for the subproblem of each individual user, the extended dual can then be solved through stochastic search with SA. In particular, we first transform the original utility maximization problem into an equivalent max–min problem. This step is based on the key observation that in the case with continuous and strictly increasing utility functions, the globally optimal solution is always on the *boundary* of the feasible (utility) region. Then, appealing to EDT and SA, we develop a distributed stochastic power control (DSPC) algorithm that stochastically searches for the optimal power allocation in the neighborhood of the feasible region’s boundary, instead of bouncing around in the entire feasible region.

Specifically, we first show that DSPC can achieve the global optimality in the underlying nonconvex optimization problem, although the convergence rate can be slow (but this is clearly due to the slow convergence nature of SA with logarithmic cooling schedule). Then, to improve the convergence rate of DSPC, we propose an enhanced DSPC (EDSPC) algorithm that employs the geometric cooling schedule[19] and performs a careful selection of penalty parameters. As a benchmark for performance evaluation, we also develop a centralized algorithm to search for the globally optimal solution over simplices that cover the feasible region. The performance gain is further verified by comparing our distributed algorithms with MAPEL[14], SEER[15], and ADP[6] algorithms. Worth noting is that the proposed DSPC and EDSPC algorithms do not require any knowledge of channel gains, which is typically needed in existing algorithms, and instead they need only the standard feedback of signal-to-interference-plus-noise (SINR) for adaptation.

Next, we integrate the proposed distributed power control approach with the back-pressure algorithm[20] and devise a joint scheduling and power allocation policy for improving the queue stability in the presence of dynamic packet arrivals and departures. This policy fits into the dynamic back-pressure and resource allocation framework and enables distributed utility maximization under stochastic packet traffic[21, 22]. Then, we generalize the study to consider multicast communications, where a single transmission may simultaneously deliver packets to multiple recipients[23, 24]. Specifically, we extend DSPC and EDSPC algorithms to multicast communications with distributed implementation, and show that these algorithms can also achieve the global optimality in terms of jointly maximizing the minimum rates on bottleneck links in different multicast groups.

The rest of the article is organized as follows. In the following section, we first introduce the system model, establish the equivalence between the utility maximization problem and its max–min form, and then develop both centralized and distributed algorithms for the max–min problem. Next, building on these power control algorithms, we develop in Section “Joint scheduling and power control for stability of queueing systems” a joint scheduling and power allocation policy to stabilize queueing systems. The generalization to multicast communications is presented in Section “Power control for multicast communications”. We conclude the article in “Conclusion” Section.