## 2 Related Literature
The authors in [9] design a distributed optimization model for joint determination of optimum power distribution and rate for SUs on a channel to sustain QoS measured using BER and minimum rate requirement. The authors in [22] analyze power and rate adaptation approach for SU capacity optimization constrained by PU interference and BER. The possibility of having many SUs sharing a channel is however ignored. A game theory approach to joint optimization of power and rate under QoS constraints is handled by [23]. The authors demonstrate that the Nash equilibrium points of the joint power and control game is actually Pareto optimal. A pricing scheme to reach the Nash equilibrium point, and hence a Pareto optimal point, is devised and used in [8]. A joint optimization problem for frequency, power and rate allocation in broadband cognitive OFDMA networks is studied in [24]. The resulting non-convex problem is solved using a greedy algorithm and a Lagrangian relaxation algorithm. The authors in [25] adopt a cross-layer strategy that jointly considers routing and MAC layer joint optimization. At the routing layer, they balance traffic by showing that sending traffic generated by each node through multiple paths significantly enables energy conservation. At the MAC layer, they assign the retry limit for retransmissions over each wireless channel. This achieves further energy conservation and improves network lifetime via load balancing. Both [26; 27] explore joint allocation of channel and transmission power but ignoring QoS requirements. Further more, the possibility of many SUs transmitting simultaneously across the channel is ignored in [26]. A joint optimization model for power and packet size in wireless sensor networks is studied in [5]. Numerical results for the resulting mixed integer programming problem are extensively explored to show the strength of joint optimization. The problem of energy optimization is re-written as a bi-objective Knapsack problem, whose solution is provided in the paper. For all the factors that have been jointly optimized, non of the earlier works studies the joint multi-objective of power, rate and delay, all considered under different QoS constraints. There is also a very rich literature on the solution methods that have been used to solve the resulting network optimization problems. The methods range from heuristics, relaxation algorithms and exact algorithms like branch and bound. [PERSON], et al. [28] propose a dynamic resource allocation problem for heterogeneous services in CRNs with imperfect channel sensing. They formulate the power and channel allocation problem as a mixed integer programming problem under constraints with the objective of maximizing total capacity of CRNs. For channel sensing, they employ the cooperatively centralized sensing method where SUs sense channels and send sensing information to the secondary Base Station (BS). This sensing method needs backbone infrastructure and has a higher cost. For a large number of users, the bandwidth for reporting is enormous. Most methods employed for spectrum sensing such as energy detection, matched filter detection and cyclo-stationary feature detection also need enormous energy for algorithmic computations. [PERSON], et al. [29] propose an energy efficient and spectrum aware multi-channel MAC protocol for cognitive radio enabled sensor networks. This is a spectrum aware asynchronous duty cycle approach that caters for the requirementsof both the CRNs and WSNs. Their scheme employs an asynchronous duty cycle approach for channel acquisition and data transmission [29]. It is a two-dimensional Markov model for changing channel conditions and queues of duty-cycled nodes with a fixed cycle length. [PERSON], et al. [30] propose an energy optimization approach (EOA) for WSNs that exploits the cross-layer principle taking into account the physical layer (transmission power), MAC layer (duty-cycling) and network layer (routing protocol).