- Open Access
Resource allocation for secondary users with chance constraints based on primary links control feedback information
© Zhou and Lok; licensee Springer. 2012
- Received: 20 December 2011
- Accepted: 10 October 2012
- Published: 21 November 2012
Resource allocation for secondary users is an important issue in cognitive radio networks. In our article, we introduce a resource allocation scheme for secondary users to share spectrum in a cognitive radio network. Secondary users can exploit the spectrum owned by primary links when the interference level does not exceed certain requirements. Uncertainties of channel gains pose a great impact on the allocation scheme. Since there are uncertainties about the channel states, we apply chance constraints to represent the interference level requirements with uncertainties. Secondary users can exceed the interference level with a predefined small probability level. Since chance constraints are generally difficult to solve and full information about the uncertain variables is not available due to the fading effects of wireless channels, we reformulate the constraints into stochastic expectation constraints. With sample average approximation method, we propose stochastic distributed learning algorithms to help secondary users satisfy the constraints with the feedback information from primary links when maximizing the utilities.
- Power control
- Cognitive radio networks
- Probabilistic constraints
- Sample average approximation
Today’s wireless networks are faced with an increase in spectrum demand while radio spectrum is a limited and valuable resource [1, 2]. On the contrary, some licensed frequency bands are unoccupied for most of the time and some other bands are just partially used while unlicensed users have needs to get access to the spectrum . Spectrum utilization can be improved significantly by allowing secondary users to exploit the spectrum unoccupied by primary users . Cognitive radio has been proposed as the method to promote the efficient utilization of the spectrum by allowing secondary users to access the spectrum when it is unoccupied [4, 5]. In cognitive radio networks, primary users may get some encouragements to provide some feedback information such that their own transmission would not be severely affected by secondary users once they want to share the spectrum . Some monitoring nodes in the networks may also assume the tasks to provide feedback information in some practical scenarios [6, 7]. The spectrum then can be better utilized while primary users’ transmissions would not be affected, which is one main motivations for cognitive radio networks. Moreover, resource allocation with time varying channels is an interesting research area in wireless communication. Uncertainties of the channel states trigger many important research issues since the uncertainties would pose an impact on the resource allocation decisions [3, 4].
Power control with spectrum sharing is an important topic in cognitive radio networks. Some researchers propose a framework for dynamic spectrum sharing in cognitive radio networks to facilitate the usage of bandwidth only under certain channel states [8, 9]. Spectrum sharing for unlicensed bands is also taken into consideration and optimal spectrum allocation is given under the condition that all channel information is available to all users . Moreover, perfect channel state information (CSI) is assumed to be known by secondary users when they have access to the spectrum [11, 12]. However, in most practical scenarios, there is an uncertainty in obtaining the channel information. Uncertainties of channel gains would affect the power allocation decisions of secondary users when they try to exploit the spectrum owned by primary users . Secondary users want to maximize their own utilities while not impairing primary users’ transmission quality even under the cases with imperfect CSI .
In some scenarios, secondary users can exceed the required interference level with some small probability instead of staying strictly below the required level. Chance constraints are widely applied in tackling uncertainties [6, 14]. Some researchers analyze the chance constraint with known distribution by reformulating into a deterministic form with some knowledge of distributions of random variables [14, 15]. Others approximate the chance constraint with convex approximation, which still requires the knowledge of the distribution even for Monte Carlo simulation . Chance constraints are generally difficult to solve especially when we do not have the full information about the distributions of the random variables due to the fading effects of wireless channels [17, 18]. Furthermore, it is difficult to obtain the exact information about the distributions in general. Sample average approximation method can be applied to help the secondary users learn the uncertain channel information [18, 19]. However, direct measurements of channel gains are difficult and transmission of such large amount of information would occupy a lot of resources. Some apply the average approximation methods in a robust way . Some consider subgradient learning method while transforming chance constraints into stochastic programming problems [21–23]. Some researchers also consider the long term ergodic optimization problems with learning processes [24, 25].
In our article, we introduce a resource allocation scheme for secondary users to share spectrum in a cognitive radio network. Secondary users can exploit the spectrum owned by primary links while their interference levels can exceed the given requirements with a small predefined probability. To tackle the chance constraints, we demonstrate a stochastic approach with sample average approximation method for secondary users to obtain the feedback information from primary links when channel gains are uncertain. Secondary users do not need to have the full information about the channel gains and they just need to know the outage information from the primary links. We propose two stochastic distributed learning algorithms (SDLA) to optimize the utilities with given feedback information.
The rest of this article is organized as follows: Section 1 introduces the system model. Section 1 illustrates deterministic approaches with chance constraints when full information about the distributions is available. Section 1 illustrates our stochastic approaches with chance constraints when full information is not available and demonstrates our SDLAs. Section 1 gives our numerical results and discussion. The last section contains our conclusions.
We consider a cognitive radio network with N primary links and M secondary users. We define the sets and . Secondary users would like to transmit to the secondary base station when accessing the spectrum. We assume that every primary link represents a channel which is orthogonal to each other. To make the notation simple, we assume that the i th primary link denotes the i th channel. Secondary users can exploit the channels when the interference caused by them would not exceed a certain level to pose a great threat for the primary link transmission. We consider the channel gains are time varying such that there are some uncertainties.
Without loss of generality, we consider the k th secondary user. We use u k (P k ) to represent the utility function for the k th secondary user. We define to represent the power vector for the k th secondary user. is the allocated power of the k th secondary user for the i th channel. In our article, we assume that u k (P k ) is a concave and monotonic increasing function with respect to every element . Utility functions with such properties are widely applied, e.g, , where is the channel gain between the k th secondary user and the secondary base station at the i th channel. The choice of utility functions is open and there are other examples of utility functions [26, 27].
P T is the total power the secondary user can use. For simplicity, we assume that all secondary users have the same total power constraint.
where V represents the predefined interference level. denotes the channel gain between the k th secondary user and the primary receiver with respect to the i th channel. To make the analysis simple, we assume all primary links have the same level requirement.
In some practical scenarios, α represents the quality of service (QoS) that the primary links demand facing interference from secondary users, e.g., when α=0, the primary links cannot tolerate any exceeded interference; when α=0.2, the primary links allow some exceeded interference with a small probability.
The interference constraints on a single secondary user may also be used in the area of admission control. In some cases, when secondary users exceed the required interference level, the primary link may block the secondary user out. Due to the uncertainty of channel gain , secondary users need to satisfy the constraints to share the spectrum. Secondary users also want to maximize their utilities while satisfying the constraints (3).
We can see that there are probabilistic constraints in the optimization problem. To obtain an optimal solution, the chance constraints themselves need to be reformulated into some deterministic forms.
Chance constraints are generally non-convex. However, in our system model, since the dimension in every chance constraint is one with respect to and , it can be shown that the constraints represent convex sets when full information about the distribution is available. Given full information about the distribution, (4) is a convex optimization problem.
Suppose that channel gain are bounded and we have the cumulative density functions (CDF) , respectively. The optimization problem is a convex optimization problem.
The constraint set is clearly convex. Thus we are maximizing a concave function with the convex set. Our optimization problem is a convex optimization problem. □
where is the maximum power for the k th secondary user at the i th channel. It can be calculated based on the CDF of the respective distribution. We also define the set Xcom=X1,com×X2,com×⋯×Xk,com×⋯×XM,com.
where is the projection into the interval and η k is a Lagrangian variable.
Under this example, secondary users can optimize power allocation with others’ choices. It can be shown that a game is formed. It can also be shown that with feasible set Xcomfor all secondary users, the game is a potential game. Thus a Nash equilibrium (NE) exists in our game.
With full information and for the game , there exists a NE in our game model.
It is obvious that the set Xcomis a convex, nonempty and compact set for all . Noting that u k is concave with respect to , the existence of NE thus follows the standard results in game theory in [28, 29]. □
With full information and for the game , it is a potential game.
for any feasible solution P k and and fixed P −k where P −k represents all other secondary users’ strategies. Therefore, Φ(P k ) is a potential function for the game the game . Due to the properties of potential game, we know that our game is a potential game . □
We propose an algorithm to help secondary users update the choices of power allocation (Algorithm 1).
3.1 Algorithm 1: Updating choice of power consumption for each user (complete information)
1: for k=1 to M do
2: Given other users’ information
3: The k th secondary user updates power allocation according to the result in (7)
4: end for
5: Until all secondary users meet the NE.
We can see that with full information about the distribution of , the result for a single secondary user is a typical water-filling solution. The algorithm is similar to iterative water filling algorithm (IWFA). The solutions and algorithms are commonly found in multiple access channels and potential games [31, 32]. Therefore, with complete information, the chance constraints in our model can be formulated into convex ones and the problem can be solved readily.
The reformulation of probabilistic constraints into deterministic convex sets requires the full knowledge of the distributions of the channel gains. Even without exact information about the distributions, we still need to measure the channel such that approximations can be applied to reformulate the probabilistic constraints. However, measurement is difficult in some scenarios.
To reduce the complexity of measurement, instead of approximating the information of the distribution for the specified random variable, we can consider an outage event as follows
When , we define that an outage occurs for the k th user at the i th channel. When , no outage occurs.
represents the random variable such that it would be one when an outage occurs and zero when there is no outage.
where denotes the j th sample of the channel gains. Equivalent, the event denotes the j th sample of the outage event based on the j th sample of the channel gain. We assume that every sample of obtained is independent and identically distributed (iid).
The expectation of the approximate equals that of the original.
4.1 Feasibility of the stochastic approximation method
When L is sufficiently large, the sample average value is used to approximate the expectation value. We have the following lemma.
This lemma says that if we take the average of L random variables to be our estimate of the expected value, then the probability of being far away from the true value is small so long as L is large.
Similarly, the whole feasible set for the sample average approximation is defined as .
where, we define the Lipschitz constant as . It can be shown that this constant can represent the bounded constant for the channel gain . and are the i th element of the vector P k and . It can be shown that the strategy set X is bounded and let D be the diameter of X. Then we have the following lemma.
where m is a finite constant and ck,maxrepresents the largest Lipschitz constant of , ∀i.
See Appendix. □
The number of primary links N is finite in all practical scenarios. We can see that by adding τ>0 and applying the sample average approximation method, the new strategy set is still feasible with large probability when the sample number L is sufficiently large.
In some practical scenarios, secondary users want to guarantee that the constraints are satisfied even with the approximation. Thus they would have γ<α and introduce the variable τ>0 to make the requirements tighter than the original ones. With these guard measures, the new strategy set is feasible with a large probability as Lemma 3 states.
and h(x) has a unique maximum at x=0.
We can see that since r is a small positive constant as defined in , when is much smaller than V, the approximate gradient is zero such that the power would increase through updating because ∇u k is positive. When is very close to V, the approximate gradient represents a negative large constant such that power may decrease. Therefore, the update scheme can help power increase while the level of is not close to V and help power decrease when the level of would be higher than V.
Moreover, since the approximate gradient is not the exact value, the initial values of power vector P k for the algorithm should be small. These kind of initial values are common in gradient update schemes and are widely applied in gradient or sub-gradient methods .
It can be shown that this approximate gradient just represents an indicator for whether subtracting λ k in the power update equations. When the expectation constraint is satisfied, the approximate gradient is zero such that power is increased through updating. Otherwise the approximate gradient is negative such that power is decreased through updating. In some practical scenarios, it can be further simplified as we will show in our simulation results.
We can see that to update the power level, secondary users need to have the feedback information about λ and the approximate gradient. The secondary user updates power levels as primal update scheme while primary links would observe the outage events and feedback the respective information through λ as dual update scheme. The result is a primal-dual update scheme in a stochastic setting.
Moreover, we also need to apply the sample average method to tackle the value of . The algorithm scheme can be summarized as follows and we denote it as SDLA-I:
Power level would be adjusted based on a gradient-like algorithm.
Step 1: Between the n th iteration and the (n + 1)th iteration, the k th secondary user obtains the feedback information of the outage event regarding the power allocation P k (n).
Step 2: The primary links calculate the outage with regard to all past history. When an outage can occur given the channel information of the j th iteration, ∀j≤n, we denote . Otherwise it is zero. We obtain an outage probability as for the n th iteration. represents the sum value for the outages that can happen during all previous iterations where we have .
Step 3: Based on the observed outage probability, the primary link would update the respective λ based on the dual update scheme and calculate the approximate gradient. The primary link would feed back such information to the k th secondary user.
Step 4: The k th secondary user would then update the power level for the (n + 1)th iteration based on the primal update scheme.
Step 5: The power level and the respective dual variable λ would be updated iteratively.
We apply learning algorithm to learn the event such that the constraint is satisfied in the long term. The secondary users can obtain some feedbacks to update the choices of power. Moreover, in the learning process, the secondary users just need to know something about the instantaneous outage events and do not need to know the distributions of specific channel gains. Also, secondary users do not need to handle the measurements. Thus the k th secondary users would just focus on the randomness of the term instead of the specified channel gains.
Chance constraints are generally difficult to tackle even when they are reformulated into expectation constraints, let alone we do not have the information about distributions. The reformulated expectation constraints are also different from general expectation constraints as discussed in stochastic programming areas, which pose great difficulties for the exact proof of convergence of the algorithms. In  and , researchers discuss the complexity of stochastic programming with chance constraints and illustrate that few analytical results can be provided even under sample average approximations, since there are some severe difficulties when facing chance constraints.
where can be obtained from previous iterations with approximations. Given a close deterministic approximations of gradient Θ k (n), the above SDLA-I can be treated as a deterministic primal-dual update algorithms. Such primal-dual algorithms have been discussed in some areas [25, 37].
In SDLA-I, though secondary users do not need to measure the channel gain , the calculation of λ k and the approximate gradient are still required in the primary links. We can further reduce the burden of such calculation such that just simple calculation and feedback information is needed for both primary links and secondary users, which is desirable in practical scenarios.
We propose another scheme to adjust power levels with observed outage probability. Instead of updating power level with information about λ and the approximate gradient, the secondary user can update the power level just with the feedback information about the observed outage probability. In this case, only primal update is realized without dual variables.
We denote it as SDLA-II. The algorithm scheme is similar to that in SDLA-I but we change some steps as follows.
Changes of Step 3: No feedback of dual variables from primary links. Secondary users check the feasibility of the constraints with the observed outage probability to see whether .
Change Step 4 as: If , secondary users regard that the constraint is satisfied and would increase the power level for the next iteration. Otherwise they regard the power is so high that it needs to be reduced to satisfy the constraint.
Changes of Step 5: There is no update of dual variables and approximate gradients. Secondary users only update the power levels iteratively.
Though the observe outage probability is not the exact true value of the expectation, it can approximate to the true value when n is sufficient large by the law of large numbers, that is, when the number of iterations is large enough.
and we define the vector form .
We can see that when the calculated outage level is smaller than α, would increase according to the general gradient algorithm. When the calculated outage level is larger than α, we regard that is so large that it should be decreased.
The secondary users update the power allocation based on the feedback of outage events such that in the long term, the probability constraints can be satisfied and the utility is maximized.
5.1 Examples of u k (.) for simulation
. Since also changes every time slot, we want to maximize the expected value. denotes the expected value with respect to G i . We shall in the sequel drop the subscript to write instead of when not leading to confusion. It can be shown that this u k (P k ) is concave and monotonic increasing with respect to every element .
There are some practical scenarios in which the channel gain would not change much. We regard that it would be a constant in the objective function. This kind of approximation is used in the stochastic modeling and algorithm in .
5.2 Simulation model
where we set ω=3.3. is generated uniformly from [20,25]. d k is generated uniformly from [1,20] m, which represents that transmissions of secondary users occur in small area. , which represents the shadowing effect, is generated from a gaussian distribution . We set generated uniformly from the interval [0.1,2] in our simulation. The value of can be varied in an interval and what we choose here represents some common environment conditions of micro-cell with shadowing effects [40, 41].
where α p is set to be 3.1. dk,p is generated uniformly from [1,15] m. κk,p=20 and is generated from a gaussian distribution . is generated uniformly from the interval [1,8] for each i and k. The large interval for represents uncertainties in these channel gains since secondary users may not be able to obtain the exact statistical knowledge of the channel gains based on the transmission environment.
We also set α=γ in our simulation. The choice of distribution of remains open and we just use this distribution to show some examples under a micro-cell with shadowing effects. For the step-size β(n) in the algorithm, when , the result may finally converge to a stable point. However, in our simulation model, to speed up the gradient-like SDLA algorithms, we set β(n) equal to one. The choice of step-size is also open. In our simulation, we just want to show some examples and try to speed up the learning process.
5.3 Simulation results and discussions
It can be seen that in Figure 1, data rates generated by SLDA-I and SLDA-II are very close to those with full information about . For both users, when the data rate is 0.6 bps/Hz, the algorithm with full information has only 0.7 dB gain over SDLA. When the data rate is 0.8 bps/Hz, the gain is just about 0.8 dB. When the total power ranges from 1 to 20 dB, the data rates under SDLA-I are very close to those under SDLA-II. When the data rate is 1 bps/Hz, for both users, SDLA-I has only 0.02 dB gain over SDLA-II. Notice that, for the algorithm with full information, optimization needs to be done every time while our SDLA can learn the result with finite iterations and do not require optimization every time.
Moreover, comparing Figure 1 with Figure 4, our SDLA algorithm can give good performance especially when the variation of is small over iterations. In Figure 4, the channel gains in the utility functions remain the same over the learning processes. The small variation of over the learning process is reasonable in some practical scenarios. For example, when the standard deviation of the shadowing effects of the channel gain is small, the distance between the transmitter and the receiver becomes the key factor such that the value of the channel gains would not change very much.
Resource allocation for secondary users with uncertainties is an important issue in cognitive radio networks. In our article, we introduce a resource allocation scheme for secondary users to share spectrum in a cognitive radio network. Secondary users can exceed the interference level with a predefined small probability level to share the spectrum. There is an uncertainty about the channel states between secondary users and primary links. We apply chance constraints to represent the interference level requirements with the uncertainties. Since chance constraints are generally difficult to solve and full information about the uncertain variable is not available, we reformulate them into stochastic expectation constraints. The secondary users can learn the outage feedback information instead of measuring certain channel gains to satisfy the constraints. We propose two SDLAs to help secondary users adjust the power to maximize the utility with only feedback information from primary links. Our simulation results show that the algorithms can give performance close to that with complete information.
Proof of Lemma 3
where A i is an event.
- Rappaport TS: Wireless Communications: Principles and Practice. Prentice Hall, New York; 2002.Google Scholar
- Proakis J: Digital Communications. McGraw-Hill, New York; 2001.Google Scholar
- Tse D, Viswanath P: Fundamentals of Wireless Communication. Cambridge University Press, Cambridge; 2005.View ArticleGoogle Scholar
- Haykin S: Cognitive radio: Brain-empowered wireless communications. IEEE J. Sel. Areas Commun 2005, 23(2):201-220.View ArticleGoogle Scholar
- Grokop L, Tse DN: Spectrum sharing between wireless networks. In Proc. 27th IEEE International Conference on Computer Communications (INFOCOM’08). Phoenix, AZ, USA; Apr 2008:201-205.Google Scholar
- Huang S, Liu X, Ding Z: Decentralized cognitive radio control based on inference from primary link control information. IEEE J. Sel. Areas Commun 2011, 29(2):394-406.View ArticleGoogle Scholar
- Zhao G, Li GY, Yang C: Proactive detection of spectrum opportunities in primary systems with power control. IEEE Trans. Wirel. Commun 2009, 8(9):4815-4823.MathSciNetView ArticleGoogle Scholar
- Acharya J, Yates RD: A framework for dynamic spectrum sharing between cognitive radios. In IEEE 2007 International Conference on Communications (ICC’07). Glasgow, Britain; June 2007:5166-5171.View ArticleGoogle Scholar
- Cendrillon R, Yu W, Moonen M, Verlinden J: Optimal multiuser spectrum balancing for digital subscriber lines. IEEE Trans. Commun 2006, 54(5):922-933.View ArticleGoogle Scholar
- Etkin R, Parekh A, Tse D: Spectrum sharing for unlicensed bands. IEEE J. Sel. Areas Commun 2007, 25(3):517-528.View ArticleGoogle Scholar
- Pang JS, Scutari G, Palomar DP, Facchinei F: Design of cognitive radio systems under temperature-interference constraints: a variational inequality approach. IEEE Trans. Signal Process 2010, 58(6):3251-3271.MathSciNetView ArticleGoogle Scholar
- Hayashi S, Luo ZQ: Spectrum management for interference-limited multiuser communication systems. IEEE Trans. Inf. Theory 2009, 55(3):1153-1175.MathSciNetView ArticleGoogle Scholar
- Xing Y, Mathur CN, Haleem MA, Chandramouli R, Subbalakshmi KP: Dynamic spectrum access with qos constraints and interference temperature. IEEE Trans. Mobile Comput 2007, 6(4):423-433.View ArticleGoogle Scholar
- Chung P-J, Du H, Gondzio J: A probabilistic constraint approach for robust transmit beamforming with imperfect channel information. IEEE Trans. Signal Process 2011, 59(6):2773-2782.MathSciNetView ArticleGoogle Scholar
- Zhang YJ, AM-C So: Optimal spectrum sharing in mimo cognitive radio networks via semidefinite programming. IEEE J. Sel. Areas Commun 2011, 29(2):362-373.View ArticleGoogle Scholar
- Nemirovski A, Shapiro A: Convex approximation of chance constrained programs. J. Opt 2006, 17(4):969-996.MathSciNetGoogle Scholar
- Ruszczynski A, Shapiro A: Stochastic Programming. In Handbooks in Operations Research and Management Science. Edited by: Shapiro A. Elsevier, Amsterdam; 2003.Google Scholar
- Robbins H, Monro S: A stochastic approximation method. Ann. Math. Stat 1951, 22: 400-407. 10.1214/aoms/1177729586MathSciNetView ArticleGoogle Scholar
- Pagnoncelli BK, Ahmed S, Shapiro A: Sample average approximation method for chance constrained programming: Theory and applications. J. Opt. Theory Appl 2009, 142(2):399-416. 10.1007/s10957-009-9523-6MathSciNetView ArticleGoogle Scholar
- Nemirovski A, Juditsky A, Lan G, Shapiro A: Robust stochastic approximation approach to stochastic programming. SIAM J. Opt 2009, 19(4):1574-1609. 10.1137/070704277MathSciNetView ArticleGoogle Scholar
- Yousefian F, Nedic A, Shanbhag UV: On stochastic gradient and subgradient methods with adaptive steplength sequences. Automatica 2012, 48(1):56-67. 10.1016/j.automatica.2011.09.043MathSciNetView ArticleGoogle Scholar
- Ram SS, Nedic A, Veeravalli VV: Distributed stochastic subgradient projection algorithms for convex optimization. J. Opt. Theory Appl 2010, 147(3):516-545. 10.1007/s10957-010-9737-7MathSciNetView ArticleGoogle Scholar
- Wang X, Gao N, access Stochasticresourceallocationoverfadingmultiple, Trans broadcastchannels: IEEE Inf. Theory. 2010, 56(5):2382-2391.View ArticleGoogle Scholar
- Ribeiro A: Ergodic stochastic optimization algorithms for wireless communication and networking. IEEE Trans. Signal Process 2010, 58(12):6369-6386.MathSciNetView ArticleGoogle Scholar
- Zhang J, Zheng D, Chiang M: The impact of stochastic noisy feedback on distributed network utility maximization. IEEE Trans. Inf. Theory 2008, 54(2):645-665.MathSciNetView ArticleGoogle Scholar
- Saraydar CU, Mandayam NB, Goodman DJ: Pricing and power control in a multicell wireless data network. IEEE J. Sel. Areas Commun 2001, 19(10):1883-1892. 10.1109/49.957304View ArticleGoogle Scholar
- Xiao M, Shroff NB, Chong EKP: A utility based power control scheme in wireless cellular systems. IEEE/ACM Trans. Netw 2003, 11(2):210-221. 10.1109/TNET.2003.810314View ArticleGoogle Scholar
- Fudenberg D, Tirole J: Game Theory. MIT Press, Cambridge; 1991.Google Scholar
- Osborne MJ, Rubinstein A: A Course in Game Theory. MIT Press, Cambridge; 1994.Google Scholar
- Monderer D, Shapley L: Potential games. Games Econ. Behav 1996, 14(1):124-143. 10.1006/game.1996.0044MathSciNetView ArticleGoogle Scholar
- Palomar DP, Chiang M: A tutorial on decomposition methods for network utility maximization. IEEE J. Sel. Areas Commun 2006, 24(8):1439-1451.View ArticleGoogle Scholar
- He G, Cottatellucci L, Debbah M: The waterfilling game theoretical framework for distributed wireless network information flow. EURASIP J. Wirel. Commun. Netw 2010, 2010: 13.Google Scholar
- Andrieu L, Cohen G, Vazquez-Abad FJ: Stochastic programming with probabilistic constraints. In 10th International Conference on Stochastic Programming. Tucson, Arizona, USA; Oct 2004:493-501.Google Scholar
- Boyd S, Vandenberghe L: Convex Optimization. Cambridge University Press, Cambridge; 2004.View ArticleGoogle Scholar
- Nemirovski A, Shapiro A: Continuous Optimization: Current Trends and Applications. Springer, Berlin, Germany; 2005.Google Scholar
- Luedtke J, Ahmed S: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Opt 2007, 19: 674-699.MathSciNetView ArticleGoogle Scholar
- Voice T: Stability of congestion control algorithms with multi-path routing and linear stochastic modeling of congestion control,. Ph.D. dissertation, University of Cambridge, 2006Google Scholar
- Bertsekas DP: Nonlinear Programming. Athena Scientific, Belmont; 2003.Google Scholar
- Bertsekas DP, Tsitsiklis JN: Parallel and Distributed Computation: Numerical Methods. Prentice Hall, Englewood Cliffs; 1989.Google Scholar
- Mark JW, Zhuang W: Wireless Communications and Networking. Edited by: Higle Julie L. Prentice Hall, Englewood Cliffs; 2003.Google Scholar
- Sklar B: Rayleigh channel in mobile digital communication systems part i: Characterization. IEEE Commun. Mag 1997, 35(7):90-100. 10.1109/35.601747View ArticleGoogle Scholar
- Branda M: Stochastic Programming E-Print Series. Reformulation of General Chance Constrained Problems Using the Penalty Functions, vol. 2, pp. 1–18, May 2010Google Scholar
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