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Abstract—We study a problem of pricing-based power al- location in collaborative primary-secondary transmission using superposition coding (SC) where ...

Pricing and Optimal Power Allocation in Collaborative Primary-Secondary Transmission Using Superposition Coding Insook Kim and Dongwoo Kim The Department of Electrical Engineering and Computer Science Hanyang University, Ansan, 426-791 South Korea Email: [email protected], [email protected]

Abstract— We study a problem of pricing-based power allocation in collaborative primary-secondary transmission using superposition coding (SC) where secondary power is assigned to primary and secondary signals with fraction 𝛼 and (1 − 𝛼), respectively. The power allocation problem for the secondary user considering price of power level is formulated as a convex optimization problem. And an optimal power allocation for a known primary’s pricing policy is provided. For the primary user, a convex optimization problem is formulated to find an optimal pricing policy. With numerical examples, optimal power allocation solutions and pricing policy for different network topologies are provided.

I. I NTRODUCTION In wireless networks, superposition coding (SC) has been applied so as to improve the region of achievable rates for collaborative users [1]. Collaborative primary-secondary transmission using SC has been considered recently in [2] and [3]. In the collaborative primary-secondary transmission protocol, a secondary transmitter applies decode-and-forward relaying scheme to transmit the primary signal along with its own (secondary) signal if the outage performance of the primary system is not affected. In [2], it is shown that a secondary transmitter within a critical radius from a primary transmitter can properly choose the fraction of transmit power to be allocated for relaying the primary signal so as to meet the outage probability requirement of the primary system and at the same time achieves secondary spectrum access. In [3], the optimal time and power allocation is investigated for a collaborative primary-secondary system. It also has been shown that optimal primary-secondary transmission protocol in terms of the achievable data rate depends on the network that consists of primary and secondary transmitter-receiver pairs. For example, collaborative primary-secondary transmission using SC increases the achievable rate when the secondary receiver is close to the secondary transmitter. In this paper, we consider pricing and optimal power allocation problem between primary and secondary signals transmitted by the secondary transmitter in collaborative primarysecondary transmission using SC. In the existing work on This work was supported by the Second-Phase Brain Korea 21 Project in 2010.

978-1-4244-6890-4/10/$26.00 ©2010 IEEE

collaborative communication using SC, the secondary user acts as a relay for the primary user with allocating a minimum fraction of its power to the primary signal so as to meet the requirement of the primary user. This could make the primary user be reluctant for the secondary user to access its licensed spectrum band. As an incentive for the primary user to gladly coexist with the secondary user, the primary user can charge the secondary user on the resource consumption in order to enhance its own revenue. The primary user can regulate the secondary user by controlling the price of the resource. Thus, pricing plays an important role in the interaction of the primary and secondary user. We model utility maximization problem for the secondary user considering price of power level used for its own signal, compared with data rate maximization problem in [3]. We provide an optimal power allocation for collaborative primary-secondary transmission for a known primary’s pricing policy. For the primary user, a convex optimization problem is formulated to find an optimal pricing policy. II. S YSTEM M ODEL We consider a wireless cognitive network that consists of two source destination pairs: (𝑆1 , 𝐷1 ) and (𝑆2 , 𝐷2 ) as in Fig. 1. We assume that there is a single channel with a normalized bandwidth of 1 Hz. (𝑆1 , 𝐷1 ) has licensed rights to operate the channel and hence we call (𝑆1 , 𝐷1 ) primary user. (𝑆2 , 𝐷2 ) can only operate on a secondary basis in the channel. Primary source 𝑆1 has radio resources such as a unit-time slot and transmit power 𝑃1 to send its own data to primary destination 𝐷1 . Secondary source 𝑆2 has not been allocated a time slot, but 𝑆2 can work as a time-duplex relay for 𝑆1 with transmit power 𝑃2 . 𝑆2 is allowed to access the time resource allocated to 𝑆1 for sending its own data to 𝐷2 only if it does not reduce the throughput of 𝑆1 . Let 𝑥1 and 𝑥2 be complex symbol vectors for the messages transmitted by 𝑆1 and 𝑆2 , respectively. Let ℎ0 , ℎ1 , ℎ2 , ℎ3 and ℎ4 be the complex channel coefficients between 𝑆1 − 𝐷1 , 𝑆1 − 𝑆2 , 𝑆1 − 𝐷2 , 𝑆2 − 𝐷1 and 𝑆2 − 𝐷2 links, respectively. The channels are assumed to be static during the following two-phase transmission. In phase 1, the source 𝑆1 broadcasts the signal 𝑥1 , and 𝑆2 and 𝐷2

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respectively. The data rates in phase 1 and phase 2 are the followings. ( ) 𝛾 1 𝑃1 (1) = 𝛿 log2 1 + 2 , (6) 𝑅𝑃 𝜎 ( ) 𝛼𝛾3 𝑃2 (2) = (1 − 𝛿) log2 1 + . (7) 𝑅𝑃 (1 − 𝛼) 𝛾3 𝑃2 + 𝜎 2 where 𝛿 and 1 − 𝛿 represents the time duration of phase 1 and phase 2, respectively. Then the end-to-end data rate of the primary data is (1)

(2)

𝑅𝑃 = min{𝑅𝑃 , 𝑅𝑃 }. Fig. 1.

simultaneously receive the signal. Of course, 𝐷1 can hear the signal but we assume that 𝐷1 ignores this signal for simplicity, which also lets no further complexity be added to the primary receiver due to the collaborative transmission. Let 𝜂𝑆2 and 𝜂𝐷2 be the noise at receiver 𝑆2 and 𝐷2 , respectively. 𝑆2 and 𝐷2 respectively receive √ (1) 𝑦 𝑆 2 = ℎ1 𝑃1 𝑥 1 + 𝜂 𝑆 2 , (1) √ (1) 𝑦 𝐷2 = ℎ2 𝑃1 𝑥 1 + 𝜂 𝐷2 , (2) where superscript (1) denotes the first phase. And then they decode 𝑥1 . 𝑆2 is going to relay 𝑥1 in the following phase 2 but 𝐷2 keeps 𝑥1 in memory. 2 the secondary source 𝑆2 transmits the signal √ In phase √ 𝛼𝑃2 𝑥𝑝1 + (1 − 𝛼) 𝑃2 𝑥2 . The signal 𝑥𝑝1 contains identical information as 𝑥1 from phase 1 but is re-encoded with a different codebook. And 𝛼 denotes the coefficient of power allocation to the primary-source signal. (√ ) √ (2) 𝑦 𝐷1 = ℎ 3 𝛼𝑃2 𝑥𝑝1 + (1 − 𝛼) 𝑃2 𝑥2 + 𝜂𝐷1 , (3) (2)

𝑦 𝐷2 = ℎ 4

(√

𝛼𝑃2 𝑥𝑝1 +



) (1 − 𝛼) 𝑃2 𝑥2 + 𝜂𝐷2 ,

We assume that 𝜂𝑆2 , 𝜂𝐷1 and 𝜂𝐷2 are the complex additive Gaussian noises with zero mean and variance 𝜎 2 . We also assume that the transmitted symbols have zero mean and unit variance. For notational convenience, let 𝛾𝑖 denote ∣ℎ𝑖 ∣2 for 𝑖 = 0, 1, ⋅ ⋅ ⋅ , 4. III. O PTIMAL P OWER A LLOCATION AND P RICING A. Problem Formulation (2)

(9)

Let 𝑅𝑃,0 denote the achievable data rate for the direct link between 𝑆1 − 𝐷1 without allocating some part of the time slot to 𝑆2 . ( ) 𝛾 0 𝑃1 . (10) 𝑅𝑃,0 = log2 1 + 2 𝜎 𝑆2 is allowable only if 𝑅𝑃 ≥ 𝑅𝑃,0 .

(11)

′ Let 𝑅𝑃 denote the data rate between 𝑆1 − 𝐷2 in phase 1. ( ) 𝛾 2 𝑃1 ′ 𝑅𝑃 = 𝛿 log2 1 + 2 . (12) 𝜎

To apply SC we have the following constraint. ′ 𝑅𝑃 ≥ 𝑅𝑃 .

(13)

In [3], they found 𝛿 and 𝛼 that maximize 𝑅𝑆 subject to (11) and (13). They showed that 𝑅𝑆 is maximized when 𝛿 = 𝛿𝐿 . ( ) log2 1 + 𝛾0 𝑃1 /𝜎 2 . (14) 𝛿𝐿 = log2 (1 + 𝛾1 𝑃1 /𝜎 2 ) (1)

(4)

where superscript (2) denotes the second phase and 𝜂𝐷1 is the noise at destination 𝐷1 . 𝐷1 decodes and gets 𝑥1 . 𝐷2 locally generates xp1 from 𝑥1 of the phase 1 signal, subtracts it from (2) 𝑦𝐷2 and finally obtains the signal √ √ (2) 𝑦𝐷2 = 𝑦𝐷2 − ℎ4 𝛼𝑃2 𝑥𝑝1 = ℎ4 (1 − 𝛼) 𝑃2 𝑥2 + 𝜂𝐷2 . (5)

(1)

Let 𝑅𝑆 denote the secondary data rate from 𝑆2 . ( ) (1 − 𝛼) 𝛾4 𝑃2 . 𝑅𝑆 = (1 − 𝛿) log2 1 + 𝜎2

System model

(8)

Let 𝑅𝑃 and 𝑅𝑃 denote the primary data rate between 𝑆1 and 𝑆2 in phase 1 and between 𝑆2 and 𝐷1 in phase 2,

𝛿𝐿 is obtained from the condition 𝑅1 ≥ 𝑅𝑃 ≥ 𝑅𝑃,0 [3]. We focus here only on determining the value of 𝛼. Therefore, we (1) (2) (2) set 𝛿 = 𝛿𝐿 and hence 𝑅𝑃 = min{𝑅𝑃 , 𝑅𝑃 } = 𝑅𝑃 in this paper, i.e., ( ) 𝛼𝛾3 𝑃2 . (15) 𝑅𝑃 = (1 − 𝛿𝐿 ) log2 1 + (1 − 𝛼) 𝛾3 𝑃2 + 𝜎 2 Suppose that the primary user charges the secondary user the amount 𝜆 per unit of transmit power for secondary-source signal during phase 2 [4]. In the proposed model, the transmit power level for the secondary-source signal is (1 − 𝛼) 𝑃2 . Therefore, the total amount charged on secondary user’s power usage (for its own signal) is 𝜆 (1 − 𝛼) 𝑃2 . For a secondary user, the payment can be seen as its cost of exploiting the primary channel. For the primary user, the payment can be seen as the compensation of its potential service quality degradation caused by partial support of secondary user. Thus the net utility of secondary user is given by

858

𝑈𝑆 = 𝜔𝑅𝑆 − 𝜆𝑃2 (1 − 𝛼).

(16)

where 𝜔 is a positive parameter which is the equivalent utility per unit data rate valuation contributing to the overall utility. Our goal is to find 𝛼 that maximize 𝑈𝑆 while keeping the constraint (11) valid. If 𝜆 is given, the design of the coefficient of power allocation 𝛼 can be posed as follows : (P-SU) max 𝛼

s.t.

𝑈𝑆 = 𝜔𝑅𝑆 − 𝜆𝑃2 (1 − 𝛼)

(17)

𝑅𝑃 ≥ 𝑅𝑃,0 , 0 ≤ 𝛼 ≤ 1.

(18) (19)

The utility of primary user can be defined as the revenue for the primary user, which is given by 𝑈𝑃 = 𝜆𝑃2 (1 − 𝛼∗ (𝜆)) ,

(20)

where 𝛼∗ (𝜆) is determined by solving the problem (P-SU) and results in a function of 𝜆, obviously. The primary user can determine 𝜆 that maximizes the revenue 𝑈𝑃 . Therefore the design of the price 𝜆 can be formulated as follows: (P-PU) max 𝜆≥0

s.t.

s.t.

max

𝑅𝑃 ≥ 𝑅𝑃,0 .

(22)

s.t.

𝑈𝑆 = 𝜔𝑅𝑆 − 𝜆𝑃2 (1 − 𝛼)

(24)

𝛼0 ≤ 𝛼 ≤ 1.

(25)

Obviously, the constraints of (P-SU) are convex functions over 𝛼. It can be easily shown that the second derivative of the objective function of (P-SU) is negative. 2

(𝛾4 𝑃2 ) ∂ 2 𝑈𝑆 𝜔 (1 − 𝛿𝐿 ) =− 2 < 0. 2 𝛼 ln 2 ((1 − 𝛼) 𝛾4 𝑃2 + 𝜎 2 )

(26)

Therefore, the objective function of (P-SU) is concave over 𝛼, and (P-SU) is a convex problem. Theorem 1: If 𝜆 is given, an optimal coefficient of power allocation for problem (P-SU) is ⎧ 𝛼 , 𝜆 ≤ 𝜆𝐿 ,  ⎨ 0 ( ) 2 ∗ 𝜔(1−𝛿 ) 1 1 𝜎 𝐿 𝛼 = (27) 1 − 𝑃2 ln 2 𝜆 − 𝛾4 , 𝜆 𝐿 ≤ 𝜆 ≤ 𝜆 𝑈 ,  ⎩ 1, 𝜆 ≥ 𝜆𝑈 , where

{

𝜆𝐿 = 𝜆𝑈 =

It is shown that an optimal coefficient of power allocation 𝛼∗ is a function of 𝜆 in Theorem 1. We discuss an optimal 𝜆 that maximizes the primary revenue in this section. (P-PU)

(21)

In (P-SU), the constraint (18) reduces to the following inequality. ) ( 𝛾 3 𝑃2 + 𝜎 2 𝜖 ≜ 𝛼0 (23) 𝛼≥ 𝛾 3 𝑃2 (𝜖 + 1) ) 1 ( (1−𝛿𝐿 ) where 𝜖 = 1 + 𝛾𝜎0 𝑃2 1 − 1. Therefore (P-SU) can be restated as (P-SU) 𝛼

C. Optimal Pricing

𝜆𝑃2 (1 − 𝛼∗ (𝜆))

B. Optimal Power Allocation

max

and 𝛼0 is in (23). Proof: See Appendix I Theorem 1 states that the following facts. 1) An optimal coefficient of power allocation 𝛼∗ is an increasing function of 𝜆. 2) Secondary power for relaying the primary signal should be allocated at least 𝛼0 so as to meet the requirement of the primary user even though relatively low price of power is provided (𝜆 ≤ 𝜆𝐿 ). 3) If the price of power for the secondary user to transmit its own signal is high (𝜆 ≥ 𝜆𝑈 ), it is not willing to allocate power to its own signal and devotes to primary signal transmission with available total power.

𝜔(1−𝛿𝐿 ) 1 ln 2 (1−𝛼0 )𝑃2 +𝜎 2 /𝛾4 , 𝜔(1−𝛿𝐿 ) 1 ln 2 𝜎 2 /𝛾4 .

𝜆≥0

𝑈𝑃 = 𝜆𝑃2 (1 − 𝛼∗ (𝜆))

(29)

𝑅𝑃 ≥ 𝑅𝑃,0 .

(30)

(P-PU) can be divided into three subproblems with respect ∗ denote to the range of 𝜆 from Theorem 1. Let 𝜆∗Λ and 𝑈𝑃,Λ an optimal 𝜆 and the primary utility in the range of Λ. 1) 𝜆 ≤ 𝜆𝐿 : We have 𝑈𝑃 = 𝜆𝑃2 (1 − 𝛼0 ), since 𝛼∗ (𝜆) = ∗ = 𝜆𝐿 𝑃2 (1 − 𝛼0 ). 𝛼0 . Therefore, 𝜆∗𝜆≤𝜆𝐿 = 𝜆𝐿 , 𝑈𝑃,{𝜆≤𝜆 𝐿} ∗ ∗ 2) 𝜆 ≥ 𝜆𝑈 : Substituting 𝛼 (𝜆) = 1 in (29), 𝑈𝑃,{𝜆≥𝜆 = 𝑈} 0. 3) 𝜆𝐿 ≤ 𝜆 ≤ 𝜆𝑈 : Let 𝛼† denote an optimal 𝛼 in the range of 𝜆𝐿 ≤ 𝜆 ≤ 𝜆𝑈 . ( ) 𝜎2 1 𝜔 (1 − 𝛿𝐿 ) 1 − 1 − 𝛼† = . (31) 𝑃2 ln 2 𝜆 𝛾4 Substituting 1 − 𝛼† in the objective function 𝑈𝑃 , 𝑈𝑃 reduces to a function of 𝜆: 𝑈𝑃,𝜆𝐿 ≤𝜆≤𝜆𝑈 =

𝜔 (1 − 𝛿𝐿 ) 𝜎 2 − 𝜆. ln 2 𝛾4

(32)

In (32), 𝑈𝑃,𝜆𝐿 ≤𝜆≤𝜆𝑈 is a strictly decreasing function over 𝜆. Therefore, we have 𝜆∗{𝜆𝐿 ≤𝜆≤𝜆𝑈 } = 𝜆𝐿 ,

(33)

𝑈𝑃 ∗{𝜆𝐿 ≤𝜆≤𝜆𝑈 } = 𝜆𝐿 𝑃2 (1 − 𝛼0 ) .

(34)

and

Finally, we have the following theorem without proof. Theorem 2: When the secondary user allocates power level to maximize the net utility in problem (P-SU), an optimal pricing policy for the primary user is

(28)

859

𝜆∗{𝜆≥0} = 𝜆𝐿 =

1 𝜔 (1 − 𝛿𝐿 ) . ln 2 (1 − 𝛼0 ) 𝑃2 + 𝜎 2 /𝛾4

(35) ■

Fig. 4. Fig. 2. The location of the communication nodes: 𝑑0 = 3, 𝑑1 = 𝑑20 , 𝑑2 = 𝑑21 , 𝑆1 = (0, 0), 𝐷1 = (𝑑0 , 0). (a) Fixed 𝛾1 , 𝛾2 , 𝛾4 and various 𝛾3 , (b) Fixed 𝛾1 , 𝛾2 , 𝛾3 and various 𝛾4 .

Fig. 3.

𝛼∗ with respect to 𝜆 : Effect of 𝛾3

It is noted that we only consider 𝛾2 ≥ 𝛾1 with the condition in (13) is met in order to apply SC. If SC cannot be applied, the data rate achieved without SC at 𝐷2 is ( ) (1 − 𝛼) 𝛾4 𝑃2 𝑅𝑆NSC = (1 − 𝛿𝐿 ) log2 1 + . (36) 𝛼𝛾4 𝑃2 + 𝜎 2 Substituting 𝑅𝑆NSC instead of 𝑅𝑆 in (16), the secondary utility function in (16) is decreasing function over 𝛼. Therefore, 𝛼NSC,∗ = 𝛼0 .

𝛾4 . The locations of 𝑆1 (0, 0), 𝐷1 (𝑑0 , 0) are fixed as in Fig. 2. In Fig. 2 (a), we consider two cases of different location of 𝑆2 and 𝐷2 : {𝑆2 , 𝐷2 } = {(𝑑0 /2, 𝜋/4), (𝑑1 /2, −𝜋/8)}, and {(𝑑0 /2, 𝜋/8), (𝑑1 /2, −𝜋/4)}. For these cases, 𝛾3 ’s are different from each case while 𝛾0 , 𝛾1 , 𝛾2 and 𝛾4 remain unchanged. In Fig. 2 (b), three cases of different location of 𝐷2 are considered: 𝐷2 = (𝑑1 /2, 𝜋/4), (𝑑1 /2, 0) and (𝑑1 /2, −𝜋/4), respectively. For the above cases, 𝛾4 varies while 𝛾0 , 𝛾1 , 𝛾2 and 𝛾3 remain unchanged. It is noted that we choose the locations of nodes in the region where the collaborative communication is promising [3]. And we assume that 𝑃1 = 𝑃2 = 10 and 𝜎 2 = 1. The parameter 𝜔 is set to 𝜔 = 1. Fig. 3 and Fig. 4 plot an optimal coefficient of power allocation 𝛼∗ for topology as in Fig. 2 (a) and Fig. 2 (b), respectively. It can be seen that 𝛼0 and 𝜆𝐿 decreases from 0.4139 to 0.2275 and from 0.1333 to 0.1112 as 𝛾3 becomes higher in Fig. 3. The variation of 𝛾3 does not effect on 𝜆𝑈 which is 0.3384 for each case. In Fig. 4, since the locations of 𝑆1 , 𝐷1 and 𝑆2 are fixed, 𝛼0 is the same for the cases, 𝛼0 = 0.4139. We also observe that 𝜆𝐿 and 𝜆𝑈 increase as 𝛾4 becomes high. When having high 𝜆𝐿 resulted from high 𝛾4 , the secondary user can hold low 𝛼 for a high 𝜆. Also the difference between 𝜆𝐿 and 𝜆𝑈 is getting higher as 𝛾4 becomes higher. This means that a response of 𝛼∗ is not sensitive for a change of 𝜆 when SNR between 𝑆2 − 𝐷2 link is high. We define a measure for an elasticity of 𝛼∗ , 𝜌=

IV. N UMERICAL E XAMPLES Let us consider the location of the communication node 𝑋 ∈ {𝑆1 , 𝐷1 , 𝑆2 , 𝐷2 } in a polar coordinate system (see Fig. 2). Let 𝑑0 , 𝑑1 , 𝑑2 , 𝑑3 and 𝑑4 be the distance between 𝑆1 − 𝐷1 , 𝑆1 −𝑆2 , 𝑆1 −𝐷2 , 𝑆2 −𝐷1 and 𝑆2 −𝐷2 links, respectively. The distance between 𝑆1 and 𝐷1 is set to 𝑑0 = 3. We assume that 𝛾𝑖 = 𝑎𝑖 /𝑑𝜈𝑖 where 𝑑𝑖 is the distance of the respective link, and 𝜈 is a path loss exponent set to 𝜈 = 4. And 𝑎𝑖 represents the effect of the other fading components. We assume that 𝑎𝑖 = 1 for all 𝑖. We consider two topologies shown in Fig. 2; (a) to investigate the effect of 𝛾3 and (b) to investigate the effect of

𝛼∗ with respect to 𝜆 : Effect of 𝛾4

1 − 𝛼0 . 𝜆𝑈 − 𝜆𝐿

(37)

High 𝜌 indicates that 𝛼∗ is elastic for a change of 𝜆. (𝜆𝐿 , 𝜆𝑈 , 𝜌) for the three cases (in descending order of 𝛾4 ) is (0.21, 4.07, 0.15), (0.18, 0.86, 0.85) and (0.09, 0.16, 8.45), respectively. Next, we plot the primary utility 𝑈𝑃 versus 𝜆 in Fig. 5 and 6. The primary utility function has a vertex at 𝜆 = 𝜆𝐿 which yields a maximum primary utility. It is seen that the maximal 𝑈𝑃∗ increases as 𝛾3 and 𝛾4 become high. This means that the primary user can charge the secondary user with the higher SNR the more amount 𝜆 as the price of transmit power for

860

Lagrangian of (P-SU) is

( ℒ(𝛼, 𝜇1 , 𝜇2 ) = 𝜔 (1 − 𝛿𝐿 ) log2 1 +

(1−𝛼)𝛾4 𝑃2 𝜎2

+𝜆𝑃2 (1 − 𝛼) − 𝜇1 (−𝛼 + 𝛼0 ) − 𝜇2 (𝛼 − 1) .

) (38)

and the gradient of ℒ(𝛼, 𝜇1 , 𝜇2 ) is

Fig. 5.

𝑈𝑃 with respect to 𝜆 : Effect of 𝛾3

Fig. 6.

𝑈𝑃 with respect to 𝜆 : Effect of 𝛾4

the secondary signal. Especially, 𝛼0 ’s are same at the peak (maximal utility) for all the cases in Fig. 5. V. C ONCLUSION In this paper, we have studied the problem of pricing-based power allocation in collaborative primary-secondary transmission using SC. The power allocation problem for the secondary user considering the price of power is formulated as a convex optimization problem and an optimal power allocation for a known primary’s pricing policy is provided. An optimal power allocation complies with SNR between 𝑆2 − 𝐷2 link. When the price for power level is cheaper than 𝜆𝐿 , the secondary user’s power allocated to relay the primary signal is 𝛼0 which is the minimum requirement to guarantee the primary user’s QoS. Meanwhile, the secondary user devotes its full power to relay the primary signal when price is higer than 𝜆𝑈 . From a view of the primary user, a problem to find an optimal pricing policy to maximize primary’s utility is formulated. We have shown that an optimal pricing policy is to charge 𝜆 = 𝜆𝐿 at which 𝛼∗ = 𝛼0 . A PPENDIX I P ROOF OF T HEOREM 1 Let 𝜇1 and 𝜇2 be the nonnegative Lagrange dual variables for the constraints 𝛼 ≥ 𝛼0 and 𝛼 ≤ 1 in (25). Then the

𝛾4 𝑃 2 𝐿) ∇𝛼 ℒ(𝛼, 𝜇1 , 𝜇2 ) = − 𝜔(1−𝛿 ln 2 (1−𝛼)𝛾4 𝑃2 +𝜎 2 + 𝜆𝑃2 + 𝜇1 − 𝜇2 . (39) Let 𝛼∗ , 𝜇∗1 and 𝜇∗2 be an optimal solution and dual variables of (P-SU), respectively. If 𝛼0 < 𝛼∗ < 1, 𝜇∗1 and 𝜇∗2 should be zero by the complementary slackness for the primal and dual inequality constraints pairs [5]. Since the gradient of ℒ vanishes at an optimal point, i.e.,∇𝛼 (𝛼∗ , 0, 0) = 0, we have the following equation for 𝛼∗ : 𝛾 4 𝑃2 𝜔 (1 − 𝛿𝐿 ) − + 𝜆𝑃2 = 0. (40) ln 2 (1 − 𝛼∗ ) 𝛾4 𝑃2 + 𝜎 2 An optimal 𝛼∗ is obtained by solving (40): ( ) 𝜎2 1 𝜔 (1 − 𝛿𝐿 ) 1 − . (41) 𝛼∗ = 1 − 𝑃2 ln 2 𝜆 𝛾4 For the existence of an optimal solution, 𝛼∗ in (41) should be in the range of 𝛼0 < 𝛼∗ < 1. Therefore, we find the following condition : 𝜔 (1 − 𝛿𝐿 ) 1 𝜔 (1 − 𝛿𝐿 ) 1

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