Joint Power Allocation and Load Balancing Optimization for Energy-Efficient Cell-Free Massive MIMO Networks

02/04/2020 ∙ by Trinh Van Chien, et al. ∙ Linköping University 0

Large-scale distributed antenna systems with many access points (APs) that serve the users by coherent joint transmission is being considered for 5G-and-beyond networks. The technology is called Cell-free Massive MIMO and can provide a more uniform service level to the users than a conventional cellular topology. For a given user set, only a subset of the APs is likely needed to satisfy the users' performance demands, particularly outside the peak traffic hours. To find a flexible and energy-efficient implementation, we minimize the total downlink power consumption at the APs, considering both the transmit powers and hardware dissipation. APs can be temporarily turned off to reduce the latter part. The formulated optimization problem is non-convex but, nevertheless, a globally optimal solution is obtained by solving a mixed-integer second-order cone program. Since the computational complexity is prohibitive for real-time implementation, we also propose two low-complexity algorithms that exploit the inherent group-sparsity and the optimized transmit powers in the problem formulation. Numerical results manifest that our optimization algorithms can greatly reduce the power consumption compared to keeping all APs turned on and only minimizing the transmit powers. Moreover, the low-complexity algorithms can effectively handle the power allocation and AP activation for large-scale networks.

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I Introduction

The use of mobile phones and other portable devices is continuously increasing the demand for data in wireless networks [2, 3]. The cellular technology has evolved over time to cater for the increasing demand but although Massive MIMO (multiple-input multiple-output) is now being used, beamforming can only mitigate the large pathloss variations in cellular deployments to a limited extent [4]. Cell-free Massive MIMO is a promising new technology to deal with the mediocre cell-edge performance by distributing the antennas over the coverage area and removing the cell edges by joint operation [5, 6, 7]

. Each distributed antenna location is called an access point (AP) and the APs transmit coherently in the downlink and process their received signals coherently in the uplink, leading to a higher signal-to-noise ratio (SNR) without using more transmit power. The coherent joint transmission in Cell-free Massive MIMO is inherited from the classical coordinated multipoint (CoMP) beamforming design with a few co-located antenna arrays

[8, 9, 10, 11, 12]

and gradually extended to scenarios with many distributed APs with few antennas. These previous network designs were mainly considering slowly fading channels where the small-scale fading realizations can be estimated perfectly and spectral efficiency (SE) utility metrics can be formulated as functions of one set of small-scale fading realizations. In contrast, a key novelty in the Cell-free Massive MIMO area is the analysis of practical fast fading channels, for which the ergodic SE is the preferred performance metric and the SE depends on imperfect channel state information (CSI) and pilot contamination. In scenarios with Rayleigh fading and some choices of linear processing, the ergodic SE can even be obtained in closed form, which makes it easier to formulate and solve practical spatial resource allocation problems.

There will be billion wirelessly connected devices by [13], which raises concerns about the power consumption and the corresponding energy-related pollution. Cellular networks have been developed to maximize the SE and coverage, leading to the norm of transmitting at the maximum allowed power in the downlink [14]. This results in high power consumption at the base stations/APs, even when the traffic is low. Upcoming technologies should be redesigned to achieve a direct connection between power consumption and traffic load [15], so that the power is low when the users request low SEs. In the context of Cell-free Massive MIMO, energy efficiency optimization has been considered in [16, 17]. These papers considered how the fronthaul power consumption can be reduced by only serving each user by a subset of the APs, but all APs are assumed to be turned on continuously. As reported in [18, 19, 20] (and references therein), the energy efficiency of heterogeneous or cloud radio access networks can be substantially improved by also turning APs on and off. However, the operating point where the energy efficiency is maximized might not provide the service quality that the users need. Hence, [21] instead considered that each user has an SE requirement that the system must satisfy with minimum power consumption, considering both the transmit power and hardware-consumed power of active APs. Hence, the goal of the resource allocation is for the system to deliver the required SEs with as low total power consumption as possible. These previous works considered cellular networks with slowly fading channels, where the channel takes one random realization throughout the entire transmission. To the best of our knowledge, there is no previous work on AP activation in Cell-free Massive MIMO networks, especially not with realistic fast fading channels where the ergodic SE must be considered. Moreover, the effects of practical aspects such as pilot contamination and different linear signal processing schemes have not been studied yet.

I-a Main Contributions

Motivated from the coexistence of multiple users using different services with stringent requirements, this paper considers that each user has a predetermined downlink SE requirement that the network must satisfy to not cause service interruption. The users and APs are arbitrarily distributed, thus it is likely that these SE requirements can be fulfilled without using all the APs. When minimizing the total power consumption in the downlink, we consider both the transmit power and the hardware-consumed power. Bearing in mind that each user will mainly be served by its neighboring APs, we consider the possibility to turn off APs that are not needed to serve the current set of users. This is an important feature since Cell-Free Massive MIMO networks may have many APs [5, 6], where the large number is needed to provide consistent coverage but might not be needed at every time instant. We formulate the new optimization problem using rigorous closed-form ergodic SE expressions for uncorrelated Rayleigh fading channels, linear precoding (either maximum ratio transmission (MRT) or full-pilot zero-forcing (F-ZF)), imperfect CSI, and pilot contamination. This allows us to optimize large-scale networks with many APs and users. The main contributions are:

  • We formulate a total downlink power minimization problem, where the active APs and transmit power allocation are the optimization variables. This problem is non-convex, but we still can obtain a globally optimal solution to both the transmit power allocation and the active APs topology by solving a mixed-integer second order cone (SOC) program.

  • Since algorithms that solve mixed-integer SOC programs are too complex for real-time applications, two heuristic low-complexity algorithms are developed by exploiting the structure of the optimization problem. The first algorithm utilizes both the optimized transmit power and sparsity, while the second algorithm only utilizes optimized transmit powers to determine which APs to turn off.

  • Numerical results demonstrate that there are scenarios where only a subset of the APs are needed to satisfy the SE requirements for all users and large power reductions can be achieved by turning off the remaining APs. Moreover, the low-complexity algorithms give total power consumptions close to the global minimum.

The rest of this paper is organized as follows: Section II gives the network model together with the downlink SE analysis. A power consumption model is introduced in Section III. Then, we formulate and solve the total power minimization problem to obtain the global optimum. Section IV propose two suboptimal algorithms with low complexity. Finally, Section V presents extensive numerical results and the main conclusions are given in Section VI.

Notations:

We use boldface lower-case and upper-case letters to denote vectors and matrices, respectively. The transpose is denoted by the superscript

and the Hermitian transpose is denoted by . The expectation operator is and

denotes a circularly symmetric complex Gaussian distribution. The Euclidean norm,

-norm, and -norm of a vector is denoted as , , and , respectively. Finally, the cardinality of the set is denoted by and represents the big- notation.

Ii System Model

We consider a Cell-free Massive MIMO network with APs and users that are arbitrarily distributed over the coverage area. A central processing unit (CPU) is connected to all APs via unlimited fronthaul links. Each AP is equipped with antennas, while there is a single antenna in each user device. We assume every user has a required SE value [b/s/Hz] that must be satisfied. At a given time instance, the users will be heterogeneously distributed and their SE requirements are likely in the interior of the capacity region of the network. Intuitively, each user will receive most of its downlink signal power from the closest APs while more distant APs typically have a negligible impact. Hence, it might suffice to only utilize a subset of the APs to satisfy the SE requirements. The remaining APs can be put into sleep mode to save power. The main goal of this paper is to find this subset and the corresponding transmit powers that satisfy the SE requirements while minimizing the total power consumption, taking the power dissipation in active APs into account.

Since the channels vary over time and frequency, the classic block fading channel model with a time division duplex (TDD) protocol is considered in this paper [22]. A coherence interval encompasses symbols and of them are dedicated to estimate the channels from uplink pilot signals. We focus on the downlink performance analysis, thus the remaining symbols are used for downlink data transmission. The channel response between AP  and user  is denoted by and is assumed to follow an independent and identically distributed Rayleigh fading model:

(1)

where denotes the large-scale fading coefficient involving both path loss and shadowing. Each channel takes an independent realization in each coherence interval. We assume the APs know the channel statistics, but the realizations need to be estimated from the uplink pilots.

Ii-a Uplink Pilot Transmission

In the uplink training phase, is a matrix gathering a set of orthonormal pilot signals that are assigned to the users. Specifically, user  transmits the pilot signal with being the pilot index. We consider a fixed and arbitrary pilot assignment but note that many algorithms have been proposed in prior work [4, 23]. We let denote the subset of users assigned to the same pilot signal as user , thus it holds that

(2)

The signal received at AP  is a superposition of the transmitted pilot signals from all the users:

(3)

where is the transmit pilot power of user  and is additive noise where each element is independently distributed as . AP  computes an estimate of from the sufficient statistics , which is obtained as

(4)
Lemma 1.

The minimum mean square error (MMSE) estimate of the channel between user  and AP  is

(5)

The channel estimate is distributed as

, in which the variance

is

(6)
Proof:

The proof is adopted from the standard MMSE estimation [24] to our notation. ∎

Lemma 1 gives the precise expression of the estimated channel between an arbitrary AP  and user . If AP  is in sleep mode, it will not estimate the channel and a convenient way to represent that is by substituting into (5), for all , which leads to a zero-valued channel estimate.

Ii-B Downlink Performance Analysis

In the downlink data transmission phase, each active AP constructs the precoding vectors based on their locally estimated channels that were computed using Lemma 1. Let us denote the precoding vector used by AP  to steer the data signal to user  as . Let denote the data symbol that is jointly transmitted to user  by all the active APs and assume . The transmitted signal at AP  to all users is

(7)

where is the transmit data power that AP  allocates to user . The received signal at user  from all the active APs is

(8)

where is independent additive noise with the zero mean and the variance . By using the capacity bounding technique described in [25, Section 2.3], [22, Section 4.3], a lower bound on the ergodic channel capacity of user  is

(9)

where , , and terms denote the desired signal, the beamforming uncertainty gain, and the inter-user interference, respectively, which are expressed as

(10)
(11)
(12)

We stress that the lower bound on the downlink channel capacity in (9) can be applied for any precoding scheme and any active AP set. To obtain closed-form expressions that can be efficiently used for optimization, we now assume the active APs either use MRT or F-ZF precoding, which are defined for as

(13)

where and is the

-th column of identity matrix

.

Lemma 2.

The downlink ergodic SE of user  is

(14)

where the effective SINR is

(15)

The parameters and depend on the selection of precoding scheme. MRT gives and . For , F-ZF gives and .

Proof:

The detailed proof aligns with [26] for MRT precoding and with [27] for F-ZF precoding, except for the different notation and that we only consider that a subset of the APs is in active mode. ∎

In (15), the numerator is proportional to , which is the array gain from the multiple antennas installed at each AP. The fact that the contributions from different APs are summed up inside the square is typical for coherent joint transmission. The first part in the denominator represents coherent interference from other users in the set , which is caused by pilot contamination. The remaining parts are the non-coherent interference and noise. If F-ZF precoding is used, each AP “sacrifices” antennas (i.e.,

spatial degrees of freedom) to cancel interference between users that have different pilots. We stress that the condition on the number of antennas

is essential for the validity of closed-form SE expression if F-ZF precoding is utilized.

The ergodic SE in (14) will hereafter be used to establish the SE constraint for each user in the network. Unlike the previous work [26, 27] that considered all APs in active mode , the new closed-form SE expressions in (14) are multivariate functions of both the transmit powers and the set of active APs. One can observe that at least a single AP should be activated, say , when the network serves users with the non-zero SE requirements. We will use these expressions to formulate and solve a new total power consumption minimization problem for Cell-free Massive MIMO networks in the next sections.

Iii Total Power Minimization Problem

To maximize the energy efficiency of the network, we can minimize the power consumption while satisfying the SE requirements of the users. This section formulates a new total power consumption minimization problem subject to transmit power constraints at the APs and the required SEs of the users. The optimization variables are the active AP set and the transmit powers. The global optimum can be found by an exhaustive search, but it is extremely costly, in particular for large networks, since the problem contains both the continuous transmit power variables and discrete variables representing the active APs. We reduce the computational complexity by transforming this non-convex problem into a mixed-integer SOC program, which is solved by the branch-and-bound approach.

Iii-a Problem Formulation

The power consumption of the network consists of both the transmit power and power dissipation in the transceiver hardware of the active APs. Similar to [28, 16], we model the total power consumption from the all active APs in the network as

(16)

where the first term in (16) is the total transmit power consumed by every active AP. The transmit power at AP  is computed as , where the scaling factor determines the inefficiency of the power amplifiers. In the second term in (16), , models the power consumption of the transceiver chain connected to active APs and the traffic-independent power of the fronthaul connections and baseband processing. In the last part of (16), (measured in Watt per bit/s) is the traffic-varying power consumption (of the fronthaul and baseband processing) that is proportional to the SE and system bandwidth  Hz.

The total power consumption minimization problem that we want to solve is

(17)

where is the maximum downlink power of AP . The SE requirement of user  is denoted as [b/s/Hz] and thus the SE in (14) must be larger or equal to this number. Note that all the transmit power variables affect all the SEs due to mutual interference.

Remark 1.

Similar optimization problems have been considered in [19, 29], but under less practical conditions. The previous optimization problems must be solved in every coherence interval since the instantaneous SE for given small scale-fading realizations are considered and the operational meaning of such an SE questionable since the codeword that fits into a coherence interval is short, while the SE concept builds on (infinitely) long codewords. Such a problem formulation limits the ability to put APs into sleep mode since each AP needs to be turned on in the pilot phase and the optimization problems must be solved very rapidly. Additionally, it has previously been assumed that perfect CSI is available everywhere, thus channel estimation and pilot reuse were not considered. In contrast, we formulate problem (17) based on the ergodic SE, which is relevant in practical networks where there is channel coding over multiple coherence intervals. This reduces the power consumption since the small-scale fading variations are exploited rather than combatted. Since the optimization depends on large-scale fading coefficients, instead of the small-scale fading coefficients, the solutions can be used over a relatively long period of time and inactive APs can be in sleep mode in the pilot transmission phase. Finally, we stress that our SE expressions take both channel estimation and pilot contamination effects into account.

In many scenarios, the network only needs to activate a subset of the APs to deliver the required SE to the users, meaning that . In order to study how many elements in are needed, we set and rewrite problem (17) with SINR constraints as

(18)

where the total hardware power consumption at AP , , is simplified from (16) based on the fact that all the SINR constraints will be achieved with equality at an optimal solution [30]:

(19)

We have reduced the computational complexity of problem (18) compared to (17) since the hardware power consumption (19) is now a constant, which transforms the objective function of problem (18) from a nonlinear to a linear function. To further simplify the problem, we introduce the notations

(20)
(21)
(22)
(23)
(24)

where are the members of the active AP set (i.e., the indices of the active APs). In (22), is defined as for MRT precoding and for F-ZF precoding. The matrix in (23) has the -th column and the -th row is denoted as . In (24), are the indices of the users belonging to the set , and is the cardinality of the set . The operator denotes the Hadamard product. We can now obtain an equivalent epigraph representation of problem (18) as

(25a)
(25b)
(25c)
(25d)

The auxiliary variable moves the objective function of problem (18) to the first constraint in (25b). We observe that for a given , problem (18) reduces to an SOC program, as previously shown in [5, 6]. Hence, although (25) is non-convex, it can be solved by making an exhaustive search over all possible selections of and solving each subproblem using convex optimization. Since at least one AP needs to be active if there is users with non-zero SE requirements, there are different selections of the APs that need to be considered in an exhaustive search. This naive approach to solving (18) will be very computationally costly even in a relatively small network.

Iii-B Globally Optimal Solution to the Total Power Minimization Problem

Instead of making an exhaustive search, a global optimum to (25) can be achieved in a structured way by utilizing, for example, using the branch-and-bound approach [31]. That would result in a more efficient implementation but the computational complexity will still grow exponentially with the number of APs. However, it enables offline benchmarking in problems with up to tens of APs and users, as will be demonstrated numerically in Section V. Let the binary optimization variable mathematically characterize the on/off activity of AP . Instead of explicitly forcing the AP’s transmit powers to zero when , we can do it implicitly by replacing its maximum transmit power by . This gives the original value when the AP is active and is zero when the AP is turned off. This feature is exploited to formulate a mixed-integer SOC program as in Lemma 3.

Lemma 3.

Consider the mixed-integer SOC program

(26a)
(26b)
(26c)
(26d)
(26e)

where is the -th row of matrix and . Moreover, the vectors and are defined as

(27)
(28)
(29)
(30)

Problems (25) and (26) are equivalent in the sense that they have the same optimal transmit powers. If we denote by

an optimal solution to the binary variables

, which is obtained by solving problem (26), the optimal set of active APs in problem (25) is

(31)
Proof:

The binary variable behaves as an indicator function which uniquely determines the activity of AP . When , the related constraint (26d) is . Since , we obtain . Alternatively, AP  will be turned off and it does not have any contribution to the total power consumption of the network as well as all terms that would have contained in the SINR expression are missing in (26c). By contrast, when , the related constraint (26d) becomes , which is a total transmit power constraint when AP  is in active mode as shown in (25). For that reason, finding results is the same as optimizing the active APs set in problem (25) by utilizing (31). ∎

The new binary variables provide the explicit link between the hardware and transmit power consumption, which is an important factor to obtain the global optimum to problem (26). A key reason that we can preserve the SOC structure, despite adding the new binary variables, is that the binary variables are not involved in the SINR constraints (26c). Instead there is an implicit connection via the zero maximum transmit power for inactive APs. This is different from the previous approaches, e.g., [18], which also defined the on/off activity using but then included it in the SINR expressions, leading to non-convex SINR constraints.

Problem (26) is a mixed-integer SOC program on standard form, thus a globally optimal solution can be obtained, for example, using CVX [32] in conjunction with the MOSEK solver [33]. This software applies the branch-and-bound approach [31] to deal with the binary variables. It is implemented in an iterative manner where the main cost of each iteration consisting three steps: finding a box, which gives a lower bound on the total power consumption, and splitting that box into the two new boxes; computing upper and lower bounds for the new generated boxes; and pruning boxes which cannot contain the optimum solution. The second step dominates the computational complexity of each iteration, while the third step decides the required number of iterations to reach the optimal solution. The following lemma provides an estimate of the computational complexity when solving problem (26).

Lemma 4.

By utilizing the standard interior-point method to solve a series of SOC programs, the computational complexity of the branch-and-bound approach to obtain a global optimum to problem (25) is in the order of

(32)

where is the accuracy of solving SOC programs along the iterations.111For a given , the set of optimized variables is called -solution to an optimization problem if the objective function at this point is at most away from the global optimum. denotes the number of iterations needed for the branch-and-bound approach to reach an optimal solution. Moreover and denote the cost of computing the lower and upper bounds (see Appendix -A for the definitions of these bounds), which are given by:

(33)
(34)

Here, and denote the number of APs already in active and sleep modes, respectively, which are obtained from the previous iteration. The initial values are . Moreover, , , and the other parameters depend on the binary indices as

Parameter
Proof:

The proof computes the computational complexity for solving SOC programs to achieve the upper and lower bounds that the branch-and-bound approach spends along iterations. A detailed derivation is provided in Appendix -A. ∎

Lemma 4 shows that the computational complexity is the total cost of computing upper and lower bounds until reaching the global optimum. Even though the computational complexity per iteration varies as the change in both the optimization variables and the procedure needed in each iteration, (33) and (34) can exhibit such features by using the big- notation. In the worst case, the branch-and-bound approach has the same computational complexity as an exhaustive search over all boxes with possible subsets of active APs. With a proper bounding rule, the average computational complexity can be significantly reduced by pruning many boxes. Nevertheless, an exponential growth with is expected. In Section V, we show that the branch-and-bound approach can find a globally optimal solution to a moderate-size network with APs.

Iv Two Suboptimal Algorithms With Lower Complexity

Motivated by the high computational complexity of solving the total power minimization problem using Lemma 3, we will now propose two algorithms that find good suboptimal solutions to problem (25) with a tolerable computational complexity and enabling implementation in large Cell-free Massive MIMO networks.

Iv-a Utilizing Sparsity to Turn Off APs

If the network does not need to turn on all the APs to provide the requested services from all the users, we know that many of the power variables will be zero. Hence, we can try to find the optimum AP subset by expressing (18) as a sparse reconstruction problem where we try to push many of the transmit power variables to become zero. To this end, we first reformulate problem (18) as a mixed -norm optimization problem.

Lemma 5.

The original problem (18) has the same optimal transmit powers as the following problem

(35)

where and each function is defined based on the transmit powers of AP  as

(36)

Moreover, if we denote by the optimal set of all transmit powers to (35), then the set

(37)

is the optimal set of active APs to problem (18).

Proof:

When AP  is in sleep mode, it assigns zero transmit power to all users (i.e., ). This AP has no contribution to the objective function of problem (35) due to and thus we can make the definition of as in (36). The optimal set of active APs is defined based on the group-sparsity structure as in (37). ∎

Lemma 5 shows that we do not need to define separate variables for optimizing the active APs set , but we can implicitly determine if AP is active or not by checking if or . The reformulated problem (35) reduces the number of optimization variables compared with (26), and in particular, all the optimization variables are now continuous. Nevertheless, problem (35) is still non-convex due to -norm in the second part of the objective function. However, we can relax -norm to an -norm for some . This is a standard relaxation technique that retains sparsity and we stress that it also gives better sparsity than an -norm relaxation (cf. Figs.  and in [34] for illustrations).222Strictly speaking, a value does not lead to norm since the subadditive property is not satisfied [35], but the “norm” terminology has anyway been used for many years and we adopt this convention. Therefore, we adopt the -norm optimization to obtain a relaxation of problem (35) as333From the range of the considered -norms, the condition as in (38) leads to .

(38)

The objective function of problem (38) treats every vector as an entity in when seeking a sparse solution. We will utilize this group-sparse property of the transmit power coefficients to solve problem (38) in a novel way, which differs from previous works that considered element-based [36] or beamforming-vector-based sparsity [21]. Even though problem (38) remains non-convex after the norm relaxation, we can find a stationary point by adapting the iteratively reweighted least squares approach from [37], that was originally developed for problems with component-wise sparsity. Specifically, after removing the exponent and the hardware power consumption in the objective function, problem (38) can be recast as

(39)

By noting that the group-sparse property implies the support of vector is the empty set, we can provide an iterative algorithm obtaining a stationary solution to problem (39).

Theorem 1.

Since the feasible set is convex, we can construct an iterative algorithm that starts with the given initial weight values and in iteration solves the SOC program

(40)

to yield the solution , for which

(41)

is the optimal transmit powers for AP  at iteration . After that, the weight values are updated for the next iteration as

(42)

where is a sufficiently small positive damping constant with . When , the proposed iterative process exhibits the properties below:

  1. The objective function of problem (39) reduces after each iteration until reaching a fixed point, which is a stationary point of problem (39).

  2. If an arbitrary AP  has zero transmit power at the optimum of iteration , this AP will have zero transmit power in all the following iterations.

Proof:

The proof is based on the convergence property of the iteratively weighted least squares approach that has been adapted to our framework. The detailed proof is available in Appendix -B. ∎

Theorem 1 guarantees a monotonically decreasing objective function and the main computational cost is to solve (40) in each iteration. The iterative process reaches a stationary point to problem (39). The second property supports turning off APs along iterations. The damping constant is introduced to cope with a numerical issue that can appear when updating the weight values (42), i.e., when . Even though the convergence properties in Theorem 1 are proved by the descending of along iterations, a sufficiently small constant value also works well in the simulations as reported in [38]. The stopping criterion can be selected by comparing two consecutive iterations. For a given accuracy , we can verify if , where is the difference of the objective function to problem (40):

(43)

The stationary point achieved by Theorem 1 may not be a globally optimal solution to problem (35) due to the norm relaxation and the inherent non-convexity. Consequently, we will not use the solution from Theorem 1 as the final solution but instead as an indication of which APs to further turn off. More precisely, we compute the transmit power that the APs utilize at the solution from Theorem 1 and reorder the APs in increasing power order.444We have implemented other possible orderings, for example, based on the total transmit power per AP or the relative maximum received power allocated to the users. For brevity, we are only considering the one that gave the best results.

Let us denote by the optimized transmit powers obtained by Theorem 1, for which a new parameter standing for the contribution of AP  is defined. Specifically, is the total received power of the users that is transmitted by AP  as:

(44)

In order to classify the contribution of each AP in

to provide the required SINRs, we define a heuristic ascending order as555Note that APs that were inactive at the solution obtained from Theorem 1 are still considered. Since the numerical precision is limited, these APs will be assigned extremely small but non-zero power, which leads to a unique ordering in (45).

(45)

where is a permutation of . We will now decide how many APs to utilize and keep only those with the largest -values using the ordering in (45). We further compute an auxiliary variable

(46)

We begin by defining a range , with the condition . Specifically, the initial values are and , then we compute the middle point at iteration as

(47)

where denotes the floor function. We now reorder the AP indices according to (45) and consider setting the first APs into sleep mode. Then, the active APs set is given by which has the cardinality . We now solve the following SOC program:

(48)

where is an upper bound defined by the sublevel set in the epigraph representation of problem (18) when the APs are in active mode. From the solution to problem (48), the new upper or lower bounds on the number of inactive APs are updated as