Based on these assumptions the joint scheduling, frequency and power allocation problem is formulated as Problem 1. All logarithms are in base 2, and equations with unspecified indices such as (3) for example, is meant to represent a set of equations which hold for all unspecified indices (such as and in (3)). The variables are defined in Table 1.

(1)

Figure 1. Visualization of a possible frequency allocation during one time-slot for an ad-hoc network of cognitive users with at most 5 available frequency bands. The arrows show a connection between a transmitter and a receiver on a specific frequency band. The narrow lines indicate a partial mesh wireless network, where a node can communicate with other nodes within its accessibility range, given the resources are available. For a node i we say a given nodes j is accessible, if.

Table 1. Algorithms variables and parameters.

(2)

(3)

(4)

(5)

where is a binary variable which indicates whether or not node is transmitting to node over frequency band and is a real variable which denotes the amount of allocated power to node on frequency. If node is not transmitting to any node on frequency, then. is the signal to interference plus noise ratio defined as

where defines transmission gain from node to node on frequency. Equation (2) corresponds to assumption 3. Equations (3) and (4) correspond to assumptions 4 and 5 for scheduling constraints. Equation (5) corresponds to assumption 6 which sets an upper bound on the power allocated to each transmitter.

3. Scheduling and Power Control in Low-SINR Regime

In contrast to our prior work [12], where Problem 1 was approached in the high-SINR regime, we aim to simplify the non-convex non-linear optimization problem in the low-SINR regime. This regime is particularly relevant in networks where the number of frequency bands is large and frequency reuse is common. The throughput maximization problem in this case is simplified by approximating the Shannon capacity with its first order Taylor series around zero SINR. That is, we approximate the objective function at low-SINR with, keeping the remaining constraints on and the same. We determine the optimum solution to this approximation to the desired problem in a number of steps:

1) In Section 3.1 we first determine a solution for the relaxation to the problem subject only to constraints on the variable domains and, and.

2) Next, in Section 3.2 we determine the solution for P matrix when the minimum SINR constraints are added,.

3) In Section 3.3 we incorporate all the scheduling constraints in (3) and (4) and show that will be binary using a suitable penalty function.

4) Finally, in Section 3.4 we introduce new variables which transform the non-linear optimization problem to a linear one.

The main conclusions to be drawn are that the optimal signaling strategy follows an on/off strategy, and that the optimization may be turned into an easily solvable linear program at the expense of increasing the number of variables from to. The core insight follows from the work of Ebrahimi et al. [10], where the authors proved that in a similar problem without frequency allocation, if the transmission power is constrained by, the power allocation strategy which maximizes, assigns either a power of 0 or to each transmitter. This is called the on/off strategy. Using a similar approach we prove that the on/off strategy also maximizes the objective function of the form subject to additional frequency and SINR constraints.

3.1. Relaxed, Unconstrained Throughput Maximization

We begin by removing most of the constraints from Problem 1 and relaxing the domains of.

**Theorem 1**. Consider the following optimization problem:

1) The optimum solution to this problem allocates either 0 or power to every node.

2) The optimum solution results in binary values for the variables.

**Proof**. A proof by contradiction was provided for the first part of the theorem by Ebrahimi et al. [10]. Using a similar argument we provide the proof for the second part. It can be seen that is convex with respect to each since (see Appendix). Thus, the maximum of will happen at the boundaries of the variable domains. To see this, assume that the optimum solution corresponds to a set with at least one non-binary element;. Since is convex with respect to, then by keeping the rest of the variables constant and moving to the boundaries (0 or 1) one can find a larger amount for and hence the set with non-binary element cannot be the optimum solution. Therefore, must hold for all.

3.2. Throughput Maximization with SINR Constraints

The only constraint in the problem addressed in previous section are on the domains of the and variables. When the SINRs are required to exceed a minimal value it is not clear if/how the optimal power allocation changes. Now we prove that adding the SINR constraint may change the set of “on” or “off” transmitters, but the on/off power allocation is still optimal. The proof is a direct result of the Generalized Lagrange Multiplier (GLM) Technique proposed by Everette [13]. Using GLM technique it can be shown that for a specific class of constrained problems, one can generate a lagrangian function of the problem such that the solution to the maximization of the lagrangian is a solution to the constrained problem. We state the needed result of [13] for clarity.

**Proposition 1**. Consider a constrained optimization problem of the form:

(6)

If the objective function of a constrained optimization problem given by Equation (6) is concave with respect to the constraint functions, then it can be guaranteed that there is a set of non-negative lagrange multipliers for the lagrangian of this problem, , such that the solution to the constrained optimization in Equation (6) and the solution to its lagrangian are identical [13].

We use this proposition directly in our throughput maximization to show that the on/off power strategy still holds when a constraint on each active link’s SINR is enforced. Since we only define the optimum solution for matrix, we assume that the matrix is given and hence SINR is only a function of. Define Problem 4:

We apply this Proposition to Problem 4 in the following theorem.

**Theorem 2.** The optimum solution to Problem 4 allocates either 0 or power to each transmitter in the network.

**Proof**. To apply Proposition 1 we must show that H is concave with respect to C, where C = []. In this problem and.

Define the set of possible solutions for. Suppose that we map every feasible matrix into a point in the space of H and C. Since is a hyperplane in the H and C space, then it is both convex and concave. Note that based on our assumption s are constants and define the slope of the hyperplane in the H and C space. Similar to the proof of Theorem 1 it can be shown that the Lagrangian corresponding to Problem 4,

is convex with respect to every. Therefore, for any non-negative set of, the solution to the unconstrained Lagrangian problem follows an on/off strategy (i.e.,). By Proposition 1, the optimal solution to the unconstrained problem is equal to that of the the constrained problem, hence the solution to Problem 4 follows an on/off strategy.

3.3. Adding Scheduling Constraints to the Optimization Problem

The last set of constraints that must be added to the problem are the scheduling constraints which are, , and. These additional constraints violate the optimum zero-one solution for the matrix. However, we can easily limit the solution to zero-one by adding a penalty function (i.e., to the objective function, where is a very large constant. This penalty function will take a very large value if any of the s has a non-binary value. Therefore the solution is directed towards a zeroone solution. Based on this discussion we state the last theorem of this section as follows:

**Theorem 3**. The solution to the nonlinear real-valued optimization problem defined as Problem 5:

subject to Equations (2)-(5)

is a zero-one solution which is also the optimum solution to this zero-one optimization problem defined as Problem 6:

subject to Equations (2)-(5)

**Proof.** For large enough the optimum solution to Problem 5 is a zero-one solution, since the penalty function added to the objective function limits the solution to only zero and one values. On the other hand the solution is a maximizer for Problem 6 as it maximizes the first term in Problem 5 for binary values of. Note that Equations (2)-(5) includes all scheduling constraints as well as the.

**Discussion**. With Theorems 2 and 3 we proved that the solution to Problem 6 with zero-one variables (i.e.,), is the same as the solution to Problem 5 with real variables. This significantly reduces the complexity of our problem from a mixed integer non-linear problem to a regular non-linear problem. With Theorem 2 we proved that the optimum solution to Problem 6 results in 0 or solutions for s. This finding helps us introduce an approach for linearizing Problem 6 in Section 3.4.

3.4. Transforming the Throughput Maximization to a Mixed Zero-One Linear Problem

The throughput maximization problem is a non-linear problem. However, in low-SINR regime when the transmitters follow an on/off power allocation strategy (based on Theorem 2), the problem can be transformed to a linear problem with the cost of additional variables. In the maximization problem both the objective function and the SINR constraint are non-linear. But both can be linearized using a variation of Balas method [14]. With the assumption of binary values for s and also 0 or values for, the objective function is transformed to:

(7)

Note that if then and so holds. Also if, regardless of the value of holds. A similar argument can be used to explain in the denominator. Equation (7) is a non-linear function of binary variables. Without loss of generality we assume that and. Also define variables

. Therefore the objective function is transformed to, with an additional constraint defining. After crossing multiplication we obtain:

(8)

Even though is not a binary variable, since it is smaller than 1, we still can use the Balas approach to linearize its multiplication with a binary variable. Note that can only take values 0 or 1 due to our scheduling constraints. Let’s define. Then Equation (8) is transformed to a linear constraint and using Balas method we only need to add a number of linear inequalities to the problem:

To transform the non-linear SINR constraint to a linear one we consider the fact that if then regardless of the value of, the constraint is satisfied. Also if then and so we can replace in the numerator of in this constraint with. We introduce additional variables to transform this constraint to a linear one. Therefore, the linear constrained optimization problem to be solved is:

subject to

This problem is a mixed zero-one linear problem with binary variables and real variables. This problem can be solved using standard methods for solving mixed-integer linear problems. Note that we did not use any approximation to transform Problem 6 (nonlinear non-convex problem) to a linear one, but the cost that we pay for this transformation is an increase in the number of variables from to. This complexity compared to the number of possible solutions in an exhaustive search which is the only reliable solution to the non-linear problems of this type is very low. The complexity of a search method for Problem 6 is

which is much higher than the complexity of our linear problem, especially for large values of N.

4. Conclusion

In this paper we presented methods for obtaining optimal solutions for throughput maximization in centralized cognitive networks operating in the low-SINR regime. The optimal solutions define the set of optimum communicating pairs as well as the optimum power levels and frequency allocated to the transmitters. We showed that in the low-SINR regime the mixed zero-one non-linear problem can be solved with an equivalent non-linear optimization problem. Also we proved that the optimal solution allocates powers to the transmitters. Using this result we transformed the mixed zero-one nonlinear problem to an equivalent mixed zero-one linear problem, which can be solved efficiently using the standard methods for solving mixed integer linear problems. The centralized optimum solution presented in this paper serves as a benchmarks for suggesting new distributed algorithms and evaluating approximation algorithms for both centralized and distributed methods.

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Appendix

Proof of.

We have. Without loss of generality we consider a given element of matrix, say, and rewrite as:

The first partial differentiation is obtained as:

The second partial differentiation is obtained as:

Since, the first term in the last equation is zero, and the second term is always positive. Therefore, always holds.