## 4 Problem Formulation
### Optimization Model  
In this sub section, we formulate a multi-objective optimization model with three objective functions, which are total network power, total network rate and total network delay. The aim of the optimization problem is to jointly maximize total network rate, minimize total network power and minimize total network delay, subject to a number of constraints. Since the channel selection variable \(\gamma_{k}^{i}\) is binary, we thus end up with a Mixed Integer Non-Linear Programming (MINP) Problem:\[\max \sum_{i=1}^{I}\sum_{k=1}^{K}\gamma_{k}^{i}R_{k}^{i}\] (17a) min \[\sum_{i=1}^{I}\sum_{k=1}^{K}\gamma_{k}^{i}p_{k}^{i}\] (17b) min \[\sum_{i=1}^{I}\sum_{k=1}^{K}\gamma_{k}^{i}\tau_{k}^{i}\] (17c) s.t. \[0\leq p_{k}^{i}\leq p_{k}^{i}\quad\forall k=1...K,\,i=1...I \tag{17d}\] \[0\leq\tau_{k}^{i}\leq\tau_{k}^{i}\quad\forall k=1...K,\,i=1...I\] (17e) \[1\leq R_{k}^{i}\leq W\log_{2}\left(1+\frac{p_{k}^{i}}{WN_{0}} \right)\quad\forall k=1...K,\,i=1...I\] (17f) \[\gamma_{k}^{i}=\{0,1\},\quad\forall k=1...K,i=1...I\] (17g) \[\sum_{k=1}^{K}p_{k}^{i}\gamma_{k}^{i}\leq\bar{P}^{i},\quad\forall i =1...I\] (17h) \[\sum_{k=1}^{K}R_{k}^{i}\gamma_{k}^{i}\leq\bar{R}^{i},\quad\forall i =1...I\] (17i) \[\tau_{k}^{i}=\frac{Ls}{R_{k}^{i}},\quad\forall k=1...K,i=1...I\] (17j) \[\sum_{i=1}^{I}\sum_{k=1}^{K}\gamma_{k}^{i}p_{k}^{i}\]0 (17k) \[\sum_{i=1}^{I}\sum_{k=1}^{K}\gamma_{k}^{i}p_{k}^{i}\]1 (17l) \[\sum_{i=1}^{I}\sum_{k=1}^{K}\gamma_{k}^{i}p_{k}^{i}\]2  
From the optimization problem (17), the total channel rate, total channel power and total channel delay are are given by Equation (17a-c), respectively. Constraints (17d) and (17e) represent the lower and upper bounds for power and transmission delay, respectively. The upper bounds for both power and transmission delay are both assumed to be constants. From constraint (17f), the rate is assumed to be bounded below by 1 and above by a function which is dependent upon the power. Constraint (17g) stands for the binary variable which has been defined in Equation (1). It is assumed that each channel has maximum allowable power and rate. From constraint (17h), the maximum allowable channel power for all users is \(\bar{P}^{i}\) and from constraint (17i), the maximum allowable rate on channel \(i\) is given as \(\bar{R}^{i}\). These upper bounds on total channel power and rate are restrictions which are normally brought about by the hardware capabilities. Constraint (17j) presents the transmission delay as a function of rate and the fixed packet data size \(Ls\)[35]. Constraints (17k) and (17l) are the QoS constraints that have been explained in Sections 4.1.1 and 4.1.2, respectively. We assume in this model that an SU is allowed to transmit on one or more channels, if conditions permit. This is captured in constraint (17m). The optimization problem (17) can be re-written as a single objective optimization by forming a weighted sum of the objective Equation (17a-c). Let \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) be the weights inserted on (17a-c), respectively, such that \(\lambda_{1}+\lambda_{2}+\lambda_{3}=1\). This technique is called _linear scalarization method_ or _weighted sum approach_[36; 37]. Suppose that the objective Equation (17a-c) are written as  
\[\sum_{i=1}^{I}\sum_{k=1}^{K}\gamma_{k}^{i}p_{k}^{i}\]3\[\sum_{i=1}^{I}\sum_{k=1}^{K}\gamma_{k}^{i}p_{k}^{i}\]4  
\[\sum_{i=1}^{I}\sum_{k=1}^{K}\gamma_{k}^{i}p_{k}^{i}\]5  
then the multi-objective optimization problem (17) can be re-written as a single objective optimization problem:  
\[\sum_{i=1}^{I}\sum_{k=1}^{K}\gamma_{k}^{i}p_{k}^{i}\]6 (21a) s.t. (17d), (17e), (17f), (17g), (17h), (17i), (17j), (17k), (17l), (17m). (21b)  
Here, \(\beta_{1},\beta_{2}\) and \(\beta_{3}\) are called normalization factors. The use of the normalization factors is to ensure that we get Pareto optimal solutions that are consistent with the weights \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\).