Abstract
We revisit the widely investigated problem of maximizing the centralized sumrate capacity in a cognitive radio network. We consider an interferencelimited multiuser multichannel environment, with a transmit sumpower constraint over all channels as well as an aggregate average interference constraint towards multiple primary users. Until very recently only suboptimal algorithms were proposed due to the inherent nonconvexity of the problem. Yet, the problem at hand has been neglected in the largescale setting (i.e., number of nodes and channels) as usually encountered in practical scenarios. To tackle this issue, we first propose an exact mathematical adaptation of the wellknown successive convex geometric programming with condensation approximations (SCVX) to better cope with large systems while keeping the convergence proof intact. Alternatively, we also propose a novel efficient lowcomplexity heuristic algorithm, ELCI. ELCI is an iterative approach, where the constraints are handled alternately based on the special property of the optimal solution, with a particular power update formulation based on the KKT conditions of the problem. In order to demonstrate ELCI’s efficiency we compare it to two stateoftheart algorithms, SCVX, and the recently proposed global optimum approach, MARL. The salient highlight of ELCI is the relatively fast and very good suboptimal performance in largescale CR systems.
Introduction
The throughput maximization in multiuser interferencelimited wireless networks has been a long standing major problem, since it is typically nonconvex due to complicated interference coupling among different links. In particular, we consider a cognitive radio (CR) system composed of secondary users (SUs) and multichannel. We aim to maximize the sumrate capacity of the secondary system, taking into account the average interference constraint at the incumbent primary users (PUs). The centralized radio resource allocation (RRA) is performed at a fusion center (FC).
Although, one of the most widely investigated problems in RRA (see e.g., [1] for an exhaustive review of related problems), only very recently few theoretically optimal algorithms were proposed [24]. In particular Qian et al. proposed one of the very first centralized algorithm MARL in [2]. Before MARL a significant number of suboptimal algorithms were proposed to handle equivalent problems. In order to circumvent the complexity of solving the initial problems, assumptions have often been made to create more tractable ones. High signaltointerferencenoiseratio (SINR) is one of the most employed assumptions (e.g., [5,6]). In the underlay CR setting [7], in addition to the classical transmit power constraint, there exists also the interference constraint (called also interference temperature) toward the primary system. To avoid the complexity of having both constraints, the authors in [8] assume the system to be in such a state that one of the two constraints becomes loose, thus can be omitted (i.e., either transmit powerlimited or primary interferencelimited system). In [9], the multiple access channel approach using successive interference cancelation yields an interferencefree formulation, thus the problem becomes convex. In particular, convex problem formulations are highly desired since the global optimum can be obtained using welldeveloped technics, e.g., the interiorpoint methods (IPM) [10]. However, those assumptions are not always valid. In particular, the opportunistic approach of the CR paradigm in addition to the interference constraint imposed by the primary system can lead to a lowSINR system with heavy interference among SUs, while some SUs can enjoy a surrounding environment free of PUs and other SUs, such that the entire system is neither in the transmit powerlimited or primary interferencelimited state (i.e., in a mix state). In contrast, other works, e.g., [2,1115] have directly handled the problems without major assumptions. Some of the most popular applied approaches are: the iterative waterfilling (e.g., [16,17]), the sequential quadratic programming (e.g., [14]), the Lagrange duality method (e.g., [18]). Many other particular iterative algorithms have been also proposed (e.g., [19]). In particular, the algorithm proposed in [11] has been considered one of the stateoftheart centralized algorithms. The algorithm (referred here as SCVX) is based on a successive convexification of the problem using the socalled condensation approximation. While, all the aforementioned algorithms are only suboptimal, the centralized optimal algorithm [2] Monotonic optimizationbAsed poweR controL (MARL), proposed by Qian et al., is based on recent advances in global optimization, including monotonic optimization and fractional programming. Later, they also proposed a distributed algorithm, aSynchrornous distributEd powEr contRol (SEER) in [20], based on the theory of Gibbs sampling. Note that the authors also proposed in [12] the MAPEL algorithm, very similar to MARL, to cope with the additional minimum rate constraints.
This said, the performance investigations have however been generally neglected in the largescale settings as it is usually encountered in practical scenarios. In particular, as also stated by the same authors in [20], MARL and MAPEL already exhibit a very slow convergence for a problem dimension of less than 8. Thus, large systems become prohibited in complexity. Alternatively, even the wellknow good suboptimal SCVX algorithm exhibits nonnegligible complexity (as later detailed), in particular with multiple channels. Therefore, although theoretically optimal, when it comes to numerical experiments, even with powerful machines, those approaches become quickly limited in terms of the problem size.
Motivated by this issue, we aim to propose algorithms with very good suboptimal performance while exhibiting good scalability. Those familiar with real numerical experiments in RRA are likely to immediately recognize that these two goals are, in general, fairly difficult to achieve together. We first propose a mathematical adaptation of the initial SCVX framework to cope with largescale settings, called LSSCVX, while preserving the SCVX convergence property [11] intact. As a second alternative, taking advantage of an important property of the problem, i.e., the optimal solution lies at the boundary of the feasible region [2], we propose an heuristic iterative algorithm called Enhanced LowComplexity Iterative approach (ELCI). The key idea of ELCI is to handle the different constraints separately, and further use a specific formulation for the iterative power updates based on the Karush–Kuhn–Tucker (KKT) conditions ([10], Section 5.5.3) of the problem. By numerical evaluations we first show that MARL, although theoretically optimal, exhibits large complexity even for small systems; Second, very similar performance to the enhanced LSSCVX can be obtained with ELCI but with much lower complexity.
It is wellknown that the main practical drawback of ELCI, as well as SCVX, MARL or MAPEL is the centralized setting with full channel state information requirement. The current trend is toward distributed approaches for practical implementation purposes and scalability of the network. It is however interesting to quantify the loss of those distributed algorithms compared to the ideal centralized sumrate case. We believe that ELCI as well as the enhanced LSSCVX can provide in practice good approximate benchmarks in the case of largescale problems as advocated by distributed algorithms. Note that although stated as an optimal distributed algorithm, the one proposed in [21] do not cope with multichannel and sumpower constraint over all bands. Similarly, in addition to the fact that the interference constraint is not considered in [20], the adaptation for multichannel and sumpower constraint is not so straightforward, in particular for largescale systems (i.e., see ([20], Eq. 13) for the multilevel integral required in the denominator).
We recently proposed an algorithm in [22] to cope with the sumrate problem. The current article builds on those results while improving the followings; First, the new algorithm has been improved for the case of multiband varying channels. Second, a lowcomplexity and optimal method is proposed to solve for the Lagrange multipliers (LGM), required in the ELCI algorithm. The method being customized for the specific need of the algorithm, the resolution exhibits much faster performance compared to using a general optimization approach (commercial tool) as in [22]. Third, only the initial SCVX provided in [11] was used in [22] such that the numerical experiments were handled with less than only 6 optimization variables. In this article, thanks to LSSCVX, we simulate with over 30 users and 128 channel bands (i.e., 3840 variables). Finally, we also consider the optimal MARL as a comparison for the small size experiments.
The remainder of this article is organized as follows: In Section 2., we define the system model. Section 3. briefly reviews the SCVX approach before describing the LSSCVX adaptation. Section 4. describes the ELCI algorithm. Section 5. reviews the key features of MARL, (LS)SCVX and ELCI. Section 6. provides numerical results to evaluate their performance, before we conclude in Section 7.
Notation: Boldface lowercase letters refer to vectors (x); boldface capital letters refer to matrices (X); single variables are referred to by simple lowercase letters (x); x_{m,n} denotes the variable in the mth row and nth column of matrix X; a constant is defined by a capital letter (X); sets are defined using script capital letters (
System model
Consider a CR network composed of a set
Figure 1. System model: cellbased uplink and adhoc underlay cognitive radio network.N BS/SU receivers, M SU transmitters, L PU pairs, and K channels.
where
We now define the main assumptions used in this work. The receiver n can estimate the instantaneous channel
We now formulate the sumrate problem for the fully synchronized CR system. Assuming that the noise and the interference have Gaussian characteristics, the network utility maximization problem is given as follows:
where the received SINR Γ_{m,k}is given in (1). The objective (2) is to allocate the resource (i.e., power and channels)
such that the sumrate capacity is maximized. Constraint (3) defines the aggregate
average interference power constraint (
This problem is known to be NPcomplete [25], even with one channel and without considering the CR constraint in Equation (3). The main problem is the interference among users. As shown in [26], if the crosstalk interference is larger than some value (see [26]), the optimal RRA is an FDMA type allocation. In fact, if the crosstalk exists but is very small (e.g., imagine very far apart set of transceivers), then the optimal solution could even be obtained using distributed approach (e.g., iterative waterfilling) to converge to a unique Nash equilibrium (NE). If the crosstalk is not negligible, multiple NE points can exist, which can be far from the social optimum.
Adapting SCVX for largescale problem, LSSCVX
In this section, we first briefly derive the convex GP form of our initial problem given by (24) to fit the successive convexification approach as in [11]. We show the exponential increase in monomial summations compared to the simplified highSINR assumption case [6][5] and one frequency band. To overcome this problem, without affecting the convergence property, we combine the approach of [11] with another convexification approach described in [27].
We start by deriving the transformation of (2)–(4) into a formulation following the standard GP form. Like linear programming (LP) or quadratic programming (QP), GP is a type of standard optimization problem [10]. The authors in [11] provide a good understanding about variations in RRA problems, their relation to GP, and possible approximations. In [27], different methods are presented to include fairness constraints. In the literature, GP deals with two types of functions: monomial and posynomial^{b}. In the standard GP problem the objective function and the inequality constraints are posynomial, and the equality constraints are monomial. In the initial optimization problem (2)–(4), the inequality constraints (3) and (4) are already in the right form. However, the initial objective function (2) requires some modifications. Maximizing (2) is equivalent to minimize the following [11]
which is not in a posynomial form due to the denominator. Directly solving this nonlinear,
nonconvex problem, and not matching any standard form, is thus difficult. Instead,
[11] proposes an approach by successively (iteratively) solving a convex approximation
problem obtained using the socalled condensation approximation method for the denominator
in (5). Let us define
where
The successive convex approximation simply repeats the GP formulation (7) and solves
it using the constraints (3), (4). Let a^{(i)}=a_{i1},…,a_{iMK}^{T},
Although the optimization variable vector in (7) has only MK elements (i.e., P), the posynomial function contains M^{MK} monomials. All the power exponents (e.g.,
Instead of applying the logsumexp on the GP expression (7) with the extremely large
monomials summation in the numerator, we apply it on (7) but using for the numerator
the multiplicative form of ψ(P) in (5), instead of the M^{MK}summations form. Taking X as the new optimization variable matrix such that
The convexity of the objective function in terms of X is due to the fact that the first element follows the logsumexp type expression which is convex ([10], p. 72), and the remaining part is linear. Note that in (8), the summations remain over MK elements. Finally, the exact same successive approach with updated condensation approximation is used as previously presented, but the more practical convex problem (8) is solved instead of (7)–(3), (4). Since the same condensation method is used, minimization in (8) is exactly equivalent to the minimization problem of the convex form of (7) with constraints (3), (4). Hence, the convergence property does not change, since each subproblem obtained from the condensation approach, being the GP convex formulation based on (7) or our proposed convex formulation (8), are to be solved optimally. The optimization problem (8) is solved using IPM, for which the gradient and Hessian derivations of the Lagrangian are relegated in the appendix. It is now possible to solve for (2)–(4) using (8) in the case of large problem, which is obviously intractable with (7). We call this approach the LSSCVX algorithm.
Enhanced lowcomplexity iterative approach, ELCI
Further motivated to obtain even lower complexity algorithms, while providing good suboptimal solutions, we present hereafter a novel heuristic algorithm, called, ELCI. ELCI is based on a similar iterative approach proposed in [28], where the authors deal with the same problem defined in (2)–(4) but without the interference constraint (3). The ELCI algorithm is designed to cope with the additional interference constraint, and also provides a fast and efficient method for solving for the required Lagrangian multipliers.
Mathematical bases
We begin by applying the KKT conditions as defined in, e.g., ([10], Section 5.5.3). However, we do not apply it to the complete set of problem given
by (2)–(4). Instead, we apply it to three different problems derived from the initial
problem: first, assume that there are no constraints (i.e., without both (3) and (4)
constraints); second, assume in addition to the objective function the interference
constraint (3) is active for a certain set of k channels and some ls of the corresponding set of
In (9) none of the constraints are taken into account; in (10) some of the interference
constraints of (3) are considered; and in (11) some of the power constraints of (4)
are considered. Let us define P^{(I)∗}P^{(II)∗}P^{(III)∗}, the optimal solutions of (9), (10), and (11), respectively, and μ_{k,l}and λ_{m} are the positive LGM. The LGMs are solved in (10) and (11), such that the respective
constraints (3) and (4) become active (e.g.,
which can be further written as
where Γ_{m,k} and I_{i,k} are defined in (1). It is easy to see that solving for P^{(I)∗}in (12) is hard since the power matrix is included in all the SINR functions, Γ. To circumvent this problem, the authors in [28] propose to use an iterative algorithm such that the power update at the next state is given by
where the relation with (12) is easy to see. Following the same approach, the power updates for the two other cases (10) and (11) are given respectively by:
The three power update expressions (13–15) form the mathematical basis of ELCI.
Solving the Lagrangian multipliers
In order to compute (14) and (15), it is first required to evaluate μ_{k,l} and λ_{m}, respectively. We present here a simple and efficient method, optimized for the particular need of ELCI. The solutions are obtained substituting (14) and (15) into (3) and (4) respectively, with equality instead of inequality constraint (i.e., becomes an active constraint).
Let us for example investigate the resolution of μ_{k,l}. For a violated PU l on channel k, μ_{k,l}is solved using (14) in (3) as follows:
where
Algorithm description
The main iterative loop of the algorithm contains three blocks A, B, and C related to the updates (13), (14), and (15), respectively. Block A updates the power without considering any of the constraints, i.e., (3) and (4). Block B updates the power, while only guaranteeing that the interference constraint (3) is satisfied for all PUs on all channels. Block C updates the power, while only guaranteeing that the transmit power constraint (4) is satisfied for all SUs.
Algorithm 1 ELCI Algorithm
Initialize: t=0,
repeat
t=t + 1
Update ∀(m,k): [Block A]
[Block B]
fork=1to Kdo
Set B_{duplicate}=OFF; Empty set
repeat
if (
if
Compute (∀m): tmp_{m}=(14) with
Update (∀m): p_{m,k}(t)=
end if
until interference constraint (3)satisfied
end for
[Block C]
Find set
for all
Compute (∀k)tmp_{k}=(15){need to computeλ_{m}}
Update(∀k):
end for
untilf(P(t))−f(P(t−1))≤ε_{0}, ε_{0}>0, (then apply an additional iteration with Finaliteration mode)
Further details about the algorithm are given as follows:
Block A: In the special case defined by the if statement, using (13) would lead to p_{m,k}(t)=∞. This stems from the fact that there is no one competing with user m on that channel k and none of the constraints (i.e., (3), (4)) are handled in (13). Thus, we constrain
to
Block B: For a given channel k, the repeat loop handles sequentially and independently the interference at the current worst
interfered PU,
Block C: The power update in Block C satisfying the power constraint (4) is handled independently from the power update in Block B satisfying the interference constraint (3). As such, power updates from Block C can violate the interference constraint (3). To make the final solution always feasible, after ELCI’s convergence criteria^{d} is satisfied, i.e., f(P(t))−f(P(t−1))≤ε_{0}, the algorithm performs an additional Finaliteration. The Finaliteration assures that all the constraints (i.e., (3) and (4)) are satisfied when terminating the ELCI algorithm thanks to Block C (i.e., through min(tmp_{k},p_{m,k}(t))).
It is interesting to note that the power updates in Blocks B and C, generally, contract the powers to fulfill the different constraints, whereas Block A expand them. Although ELCI separates the interferencelimited update (i.e., (14)) and the sumpowerlimited update (i.e., (15)) cases, the iterative method presented in Algorithm 1 basically aims to recombine (yet nonoptimal) the two independent updates. Thus, ELCI does not only aim one of the two system cases, as opposed to, e.g., [8].
Performance and complexity
MARL
Although the problem in (2)–(4) is nonconvex, three important features enable MARL
to optimally solve it, assuming infinite time [2]. One of those important features is the monotonically increasing objective function
in terms of SINR. The MARL algorithm constructs a sequence of polyblocks to approximate
the SINR region boundary with an increasing level of accuracy. By tuning an error
tolerance parameter, a tradeoff between performance and convergence time can also
be achieved. However, as explained in [12] the polyblock vertices projection approach does not exploit the shape of the feasible
region, but shrink from every side in the continuous
Figure 2. Close sumrate performance illustration between MARL, LSSCVX and ELCI, for a small CB system. Two adjacent cells of 100 m radius with one SU each (i.e., M=N=2), two channels (i.e., K=2), and one PU pair (i.e., L=1). Cellbased simulation using 100 random channel gain realizations.
Figure 3. Illustration of MARL’s slow convergence for one random small scenario realization. Four adjacent cells of 100 m radius with one SU each (i.e., M=N=4), two channels (i.e., K=2), and one PU (i.e., L=1).
SCVX/LSSCVX
Although the convexification approach in [11] has been generally considered to be quite efficient, it does not necessarily converge to the globally optimal solution(s) due to the condensation approximations. An improper initialization may also impact its optimality performance [2]. Although not addressed in this current article, [30] provides some efficient methods for guessing good initial points. The barrier method (classical IPM), used in this article to solve the convex optimization (8), has the nice property that the number of Newton steps grows very slowly with the problem dimension [10]. However, the computational effort to carry out one Newton step also grows with the dimension. As explained in [31], in general, when exact second derivatives can be computed with reasonable computational effort, it is usually a good idea to use them, since the IPM normally converges in fewer iterations and is more robust. When the problem has a dense Hessian or nonsparse Hessian, using the quasiNewton approximation might be better, even if the number of iterations increases, since the computation time per iteration for the search direction using the exact derivative might be significantly higher. However, since we noticed some losses with the quasiNewton (BFGS) compared to the IPM, we use the latter approach in our simulations where the Hessian for the Lagrangian of (8) is provided as appendix.
ELCI
The suboptimality of ELCI is mainly due to the fact that the constraints (3) and
(4) are taken into account in alternation instead of simultaneously. In order to circumvent
possible local optima, most of the heuristics algorithms proposed in the literature
can repeat the locallyoptimal algorithm with different random initial values. However,
like MARL, we always initialize
Although ELCI is not proved to necessarily converge, extensive simulations (i.e., some presented in Section 6.) indicate that the algorithm, always converges, or approximately converges with continuous slight oscillation^{d}. As later shown, comparing ELCI to LSSCVX algorithm no divergence of the former algorithm has been observed.
Numerical experiments
This section compares through various numerical experiments the performance of ELCI against LSSCVX.
We define the channel gain such that
In Figures 4 and 5 we set M=20and K=25and provide the PDF (probability density function) of their performance ratio (i.e.,
Figure 4. PDF of the sumrate ratio between ELCI and LSSCVX’s first trial, for averagescale scenarios. Three settings with M=20and K=25: 1) cellbased (N=4), 2) adhoc (N=M), and 3) cellbased with nonvarying (referred as n. v.) channels (i.e., same channel gain over all K channels for a given link is enforced). Each PDF is based on 500 random realizations.
Figure 7 illustrates the average sumrate performance of ELCI and LSSCVX for various largescale settings. The corresponding time consumption is provided in Figure 8. Each point has been averaged over 50 realizations. While the performance of ELCI in Figure 7 is equal or very close to LSSCVX (i.e., in fact in the loglog scale plot of Figure 7, the difference is almost not visible), the much lower complexity of ELCI is clear from Figure 8. Again, due to more complicated coupling among different links for the adhoc system as K becomes very large ELCI has more loss of performance compared to the CB case, but again still minor.
Figure 7. Close sumrate performance illustration between LSSCVX and ELCI for various largescale settings, in both CB and adhoc networks. Each point is averaged over 50 random realizations.
Conclusion
The problem of interferencelimited sumrate capacity in wireless network has been one of the most widely investigated problem, with and without cognitive radio constraint. Yet, only very recently centralized algorithms like MARL have been proposed yielding the optimal solution. However, in practical numerical experiments MARL cannot cope with large problems. Motivated by this issue, we propose two alternatives. First, we propose a clear mathematical adaptation LSSCVX of the wellknown stateoftheart suboptimal algorithm, SCVX, to cope with large problems, while keeping the initial convergence proof unchanged. Second, as a further alternative, we propose a new lowcomplexity heuristic algorithm, ELCI, based on the fact that the optimal solution lies on the boundary of the feasible region. The key idea of ELCI is to handle the different constraints separately, and further use a specific formulation for the iterative power updates based on the KKT conditions of the problem. Compared to LSSCVX, ELCI was shown to provide an excellent tradeoff for very largescale system applications where both good performance and low complexity is required.
Appendix 1
The Lagrangian function for the optimization problem (8) can be written as
where Φ(X) is the objective function (8), and μ_{k,l},λ_{m}≥0are the Lagrange multipliers related to the constraints (3) and (4), respectively. The gradient and Hessian of the Lagrangian function are given as follows:
Endnotes
^{a} Simple attribution in terms of the highest channel power. In case each subchannel have different gains, a SU selects the BS with the best channel averaged over all subchannels.
^{b}
^{c} IPM is one of the most popular generic approach for solving any convex problems, but one can also use other generic methods, e.g., cuttingplane method.
^{d} When ε_{0}is too small, a continuous small oscillation may appear, and the convergence criteria (i.e., f(P(t))−f(P(t−1))≤ε_{0}) cannot be satisfied. In such a case it suffices to terminate the algorithm and apply the Finaliteration.
Competing interests
The authors declare that they have no competing interests.
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