A Characterization of Max-Min SIR-Balanced Power Allocation with Applications
Sławomir Stańczak, Michał Kaliszan, Nicholas Bambos, Marcin Wiczanowski
aa r X i v : . [ c s . I T ] J u l A Characterization of Max-Min SIR-BalancedPower Allocation with Applications
Sławomir Sta ´nczak † , Michał Kaliszan † , Nicholas Bambos ∗ and MarcinWiczanowski †† Fraunhofer German-Sino Lab for Mobile CommunicationsEinsteinufer 37, D-10587 Berlin, GermanyEmail: { stanczak, michal.kaliszan, marcin.wiczanowski } @hhi.fraunhofer.de ∗ Department of Electrical Engineering350 Serra Mall, Stanford University, Stanford, CA 94305Email: [email protected]
Abstract
We consider a power-controlled wireless network with an established network topology in whichthe communication links (transmitter-receiver pairs) are corrupted by the co-channel interference andbackground noise. We have fairly general power constraints since the vector of transmit powers isconfined to belong to an arbitrary convex polytope. The interference is completely determined by a so-called gain matrix. Assuming irreducibility of this gain matrix, we provide an elegant characterization ofthe max-min SIR-balanced power allocation under such general power constraints. This characterizationgives rise to two types of algorithms for computing the max-min SIR-balanced power allocation. Oneof the algorithms is a utility-based power control algorithm to maximize a weighted sum of the utilitiesof the link SIRs. Our results show how to choose the weight vector and utility function so that theutility-based solution is equal to the solution of the max-min SIR-balancing problem. The algorithm isnot amenable to distributed implementation as the weights are global variables. In order to mitigate theproblem of computing the weight vector in distributed wireless networks, we point out a saddle pointcharacterization of the Perron root of some extended gain matrices and discuss how this characterizationcan be used in the design of algorithms in which each link iteratively updates its weight vector in parallelto the power control recursion. Finally, the paper provides a basis for the development of distributed power control and beamforming algorithms to find a global solution of the max-min SIR-balancingproblem.
Index Terms
Max-min SIR-balancing, Max-min fairness, Power control, Wireless networks, Utility maximization,Interference management, Distributed algorithms
I. I
NTRODUCTION
Wireless channel is error-prone and highly unreliable being subject to several impairmentfactors that are of transient nature, such as those caused by co-channel interference or multipaths.Excessive interference can significantly deteriorate the network performance and waste scarcecommunication resources. For this reason, strategies for resource allocation and interferencemanagement are usually necessary in wireless networks to provide acceptable QoS levels to theusers.There are different mechanisms for resource allocation and interference management. Powercontrol may play a central role in distributed wireless mesh networks, where, due to the lackof a central network controller, link scheduling strategies are notoriously difficult to implement.Thus, a reasonable approach is to avoid only strong interference from neighboring links, andthen use an appropriate power control policy to manage the remaining interference in a network.In this paper, we focus on the power control problem, which addresses the issue of coordinatingtransmit powers of links such that the worst signal-to-interference ratio (SIR) balanced againstsome SIR targets attains its maximum. This so-called max-min SIR-balancing problem is awidely studied resource allocation problem for wireless networks (see, for instance, [1], [2], [3],[4], [5], [6], [7], [8], [9], [10], [11], [12] as well as [13, Sections 3.1 and 5.6], [14], [15, Section5.6]). A key feature of this strategy is that any given SIR (signal-to-interference ratio) targetsare feasible if and only if they are satisfied under a max-min SIR-balanced power allocation.Moreover, the notion of max-min SIR-balancing is closely related to max-min fairness, themost common notion of fairness. Note that since we focus on static wireless networks, transmitpowers are to be periodically adjusted to changing channel and network conditions (dynamicpower control). This in turn presumes a relatively low up to moderate network dynamics. Incontrast, in highly dynamic wireless networks, one should consider resource allocation schemes for stochastic wireless networks [16], [17], [18], [19], [20], [21], [22], [23].In the noiseless case, which is widely considered in the literature [1], [2], [3], [5], [4], [24],[25], [26]. (an overview can be found in [11], [27]) and where power constraints play no rolein the analysis, it is widely known [11], [13, Sections 3.1] that any positive eigenvector of the(irreducible) gain matrix scaled by a diagonal matrix of given SIR targets is a solution to themax-min SIR-balancing problem. In [12], the problem was solved for a “noisy” downlink channelconstrained on total power. The sum constraint on transmit powers was captured by an additionalequation so that the optimal solution is characterized in terms of some unique eigenvector of acertain irreducible gain matrix of higher dimension (see also [13, pp. 111-113]).Assuming an irreducible gain matrix, Sections III–IV extend these results to any convexpolytope as the constraint set to model constraints on transmit powers of the links. In additionto the analysis in a higher dimension (as in [12]), we also obtain an elegant characterizationof the max-min SIR-balanced power vector from an eigenvalue problem of the same dimensionas the original problem. These results were inspired by [28], where the authors used differenttools to characterize rate region in interference channel with constrained power (see also theacknowledgments after the conclusions at the end of this paper).In Section V, we use the results of Sections III–IV to establish a connection between themax-min SIR-balancing power control problem and the utility-based power control problem.Such a connection is known in the noiseless case [15, Section 5.9] and constitutes the startingpoint for the analysis in [14]. More precisely, we show how to choose the weight vector andutility function so that solving the problem of maximizing a weighted sum of the utilities ofthe SIRs leads to the max-min SIR-balancing solution. This result was used in [29] to solve themax-min SIR-balancing problem over the joint space of admissible power vectors and receivebeamformers. Thus, the results of this paper provide key tools to generalize the results of [14]to noisy channels under general power constraints and to a larger class of utility functions.An advantage of the utility-based approach is that there exist distributed power control schemesto compute the max-min SIR power vector, provided that each link knows how to select its weight[30], [31]. The problem is however that a desired weight vector is determined by positive leftand right eigenvectors of some nonnegative matrix, so that the links cannot choose their weightsindependently. Thus, as neither the eigenvectors nor the corresponding matrices are a prioriknown at any node, the presented approach for computing the max-min SIR power allocation is still not amenable to implementation in decentralized wireless networks.A basic idea to overcome or at least to alleviate this problem is to let each link iterativelyupdate its weight vector in parallel to the power control recursion. In Section V-B, we pointout that a saddle point characterization of the Perron roots of some nonnegative matrices. Thischaracterization provides a basis for efficient saddle point algorithms converging to a max-min SIR-balanced power vector. Basically, the idea is redolent of primal-dual algorithms thatemploy some optimization methods to minimize the Lagrangian over primal variables and to simultaneously maximize it over dual variables.Before starting with the analysis, we introduce definitions, notation and state the max-minSIR-balancing problem. II. D
EFINITIONS AND PROBLEM STATEMENT
We consider a wireless network with an established network topology, in which all links sharea common wireless spectrum. Let K ≥ users (logical links) compete for access to the wirelesslinks and let K = { , . . . , K } denote the set of all users. The transmit powers p k , k ∈ K , of theusers are collected in the vector p = ( p , . . . , p K ) ≥ , which is referred to as power vector or power allocation . The transmit powers are subject to power constraints so that p ∈ P where P = { p ∈ R K + : Cp ≤ ˆ p , C ∈ { , } N × K } ⊂ R K (1)for some given ˆ p = ( P , . . . , P N ) > and C with at least one in each column so that P is acompact set. Throughout the paper, we use N = { , . . . , N } where N is the number of powerconstraints. The main figure of merit is the SIR at the output of each receiver given by(A.1) SIR k ( p ) = p k /I k ( p ) , k ∈ K , where the interference function I k is I k ( p ) = ( Vp + z ) k = P Kl =1 v k,l p l + z k . V := ( v k,l ) ∈ R K × K + is the gain matrix , v k,l = V k,l /V k,k if l = k and if l = k where V k,l ≥ with V k,k > is the attenuation of the power from transmitter l to receiver k . The k th entry of z := ( z , . . . , z K ) is z k = σ k /V k,k , where σ k > is the noise variance at the receiver output.Let γ , . . . , γ K > be the SIR targets and let Γ = diag( γ , . . . , γ K ) . We say that the SIRtargets are feasible if there exists a power vector p ∈ P (called a valid power vector) such that SIR k ( p ) ≥ γ k > . R + , R ++ are nonnegative and positive reals, respectively. Definition 1:
Given any Γ , ¯ p is said to be a max-min SIR-balanced power vector if ¯ p := arg max p ∈ P min k ∈ K (SIR k ( p ) /γ k ) . (2)It is important to notice that the maximum in (2) exists as min k ∈ K (SIR k ( p ) /γ k ) is continuouson the compact set P . Thus, γ k are not necessarily met under optimal power control (2). Forthis reason, γ k can also be interpreted as a desired SIR value of link k . A trivial but importantobservation is that ¯ p > , allowing us to focus on P + = P ∩ R K ++ .III. S OME P RELIMINARY O BSERVATIONS
In general, ¯ p of Definition 1 is not unique. For general power constraints, the uniqueness isensured if V is irreducible [32], [33] since then the links are mutually dependent through theinterference. In order to see this, notice that the problem (2) is equivalent to finding the largestpositive threshold t such that t ≤ SIR k ( p ) /γ k for all k ∈ K and p ∈ P . The constraints canbe equivalently written in a matrix form as Γz ≤ ( t I − ΓV ) p and p ∈ P . So, as p must be apositive vector, [15, Theorem A.51] implies that the threshold t must satisfy ρ ( ΓV ) < /t . (3)Now, one particular solution to the max-min SIR-balancing problem (2) is ¯ p ′ given by ¯ p ′ = (cid:0) /t ′ I − ΓV (cid:1) − Γz , ¯ p ′ ∈ P (4)where t ′ = arg max t ≥ t s.t. (1 /t I − ΓV ) p = Γz , p ∈ P . (5)Note that ¯ p ′ is a max-min SIR-balanced power vector such that SIR k (¯ p ′ ) /γ k = SIR l (¯ p ′ ) /γ l foreach k, l ∈ K . This immediately follows from (4) when it is written as a system of K SIRequations. By (3), (4), (5) and [15, Theorem A.51], ¯ p ′ > exists and is a unique power vectorcorresponding to a point in the feasible SIR region that is farthest from the origin in a directionof the unit vector γ / k γ k .The above considerations are illustrated in Fig. 1. The plots depict two examples of feasibleSIR regions in a system with two users. In both cases, the point (SIR(¯ p ′ ) , SIR(¯ p ′ ) ) , with ¯ p ′ The feasible SIR region is the subset of R K + of all SIR levels that can be achieved by means of power control. F defined by(7) becomes the feasible SIR region if φ ( x ) = x and γ k = 1 , k ∈ K . SIR SIR γ SIR γ SIR (¯ γ ′ , ¯ γ ′ ) (¯ γ , ¯ γ ) ( γ ′′ , γ ′′ ) (¯ γ ′ , ¯ γ ′ ) = ( γ ′′ , γ ′′ ) Fig. 1. The feasible SIR region under individual power constraints and two different gain matrices V ≥ . The followingnotation is used: ¯ γ k = SIR k (¯ p ) and ¯ γ ′ k = SIR k (¯ p ′ ) where ¯ p and ¯ p ′ are defined by (2) and (4), respectively. Left: V is chosenso that SIR ( p ) = p /z , in which case ¯ p is not unique. Right: V is irreducible, in which case ¯ p is unique and equal to (4).The point ( γ ′′ , γ ′′ ) corresponds to the max-min fair power allocation. given by (4) is the point at the intersection of the boundary of the feasible SIR region withthe line defined by the γ vector and is max-min SIR-balanced. It is however not the uniquesolution if SIRs of some users can be increased without affecting the minimum. The examplesof configurations in which the power allocation ¯ p ′ given by (4) is not and is a unique solutionof the max-min SIR-balancing problem are presented in the left and in the right subplot of Fig.1, respectively.Now let us assume throughout the paper that(A.2) φ : R ++ → Q ⊆ R is continuously differentiable and strictly increasing function.By strict monotonicity, one has φ (min k ∈ K SIR k ( p ) /γ k ) = min k ∈ K φ (SIR k ( p ) /γ k ) for every p > . Thus, as ¯ p ∈ P + , ¯ p = arg max p ∈ P + min k ∈ K φ (SIR k ( p ) /γ k ) (6)where ¯ p is a max-min SIR-balanced power vector defined by (2). With (6) in hand, we canprove a sufficient condition for the uniqueness of ¯ p under general power constraints. To thisend, given φ , we define the set F ⊂ Q K as F = (cid:8) q ∈ Q K : q k = φ (cid:0) SIR k ( p ) /γ k (cid:1) , k ∈ K , p ∈ P + (cid:9) (7)Note that F can be interpreted as a feasible QoS region where the QoS value for link k is φ (SIR k ( p ) /γ k ) . By (A.2) and [15, Sect. 5.3], the following can be said about F . Observation 1:
There is a bijective continuous map from F onto P + (see also (23)). If The reader should bear in mind that the existence of such a bijective map allows us to prove some results in F . (A.3) the inverse function g ( x ) := φ − ( x ) is log-convex [15],then F is downward comprehensive, connected and convex . Moreover, SIR k (¯ p ′ ) with ¯ p ′ definedby (4) corresponds to a point on the boundary of F where q = . . . = q K with q k defined by (7).Note that the boundary of F (denoted by ∂ F ) is the set of all points of F such that, if p is the corresponding power vector in (7), then Cp ≤ ˆ p holds with at least one equality.Widely known examples of functions satisfying (A.2) and (A.3) are x log( x ) , x > , and x
7→ − /x n , n ≥ , x > . Now let us state the following auxiliary result. Lemma 1:
Suppose that (A.2) and (A.3) hold. Let V be irreducible. Then, q ∈ ∂ F if andonly if there exists w > such that q maximizes x w T x over F . Proof:
By convexity and downward comprehensivity of F , every boundary point of this setis a maximal point and it maximizes x w T x over F for some w ≥ [34, pp. 54–58]. Since V is irreducible, it follows from [35, Theorem 4.3] (see also [35, Corollary 4.3]) that w > .Now we can easily observe the following. Observation 2: If V ≥ is irreducible, then ¯ p is unique and equal to ¯ p ′ defined by (4). Proof:
The proof is deferred to Appendix VIII-A.We complete this section by pointing out a connection between the max-min SIR-balancingproblem considered in this paper and the notion of max-min fairness. A vector of balancedSIRs with entries
SIR k ( p ′′ ) /γ k , k ∈ K is max-min fair if any SIR k ( p ′′ ) /γ k cannot be increasedwithout decreasing some SIR l ( p ′′ ) /γ l , l = k which is smaller than or equal to SIR k ( p ′′ ) /γ k ; thevector p ′′ is then a max-min fair power allocation [36], [37]. A max-min fair power allocationis therefore also max-min SIR-balanced (provided that the feasible SIR region is downwardcomprehensive); the converse is in general not true. However, if the max-min SIR-balancingproblem has a unique solution given by (4), this solution is also max-min fair and there areno other max-min fair power allocations. These relations can be observed in Fig. 1 where bothmax-min SIR-balanced and max-min fair points are indicated.IV. C HARACTERIZATION OF M AX -M IN SIR-
BALANCING
In this section, we characterize ¯ p ∈ P + defined by (2) under the assumption that V isirreducible. We point out possible extensions to reducible matrices at the end of this section.We assume that P ⊂ R K + is a convex polytope given by (1). So, throughout this section, max n ∈ N g n ( p ) ≤ where g n ( p ) := 1 /P n c Tn p , n ∈ N (8)and c n ∈ { , } K is a (column) vector equal to the n th row of the matrix C . Now, using (8),the max-min SIR power vector ¯ p defined by (2) can be written as ¯ p = arg max p ≥ min k ∈ K (SIR k ( p ) /γ k ) s.t. max n ∈ N g n ( p ) ≤ . (9)where γ k > , k ∈ K , is arbitrary but fixed. Lemma 2:
Let ¯ p be any power vector that solves (9). Then, the following holds(i) max n ∈ N g n (¯ p ) = 1 .(ii) If V ≥ is irreducible, then ¯ p is unique and ∀ k ∈ K γ k / SIR k (¯ p ) = β, β > . (10) Proof:
The proof can be found in Appendix VIII-B.Because ¯ p maximizes min k ∈ K (SIR k ( p ) /γ k ) over P , it follows from (10) that /β > is thecorresponding maximum. It must be emphasized that (10) is not true for general nonnegativematrices V ≥ . In the lemma, we require that the gain matrix be irreducible, which is sufficientfor (10) to hold but not necessary. The irreducibility property ensures that, regardless of thechoice of P , there is no subnetwork being completely decoupled from the rest of the network.To be more precise, if V is irreducible, then the network is entirely coupled by interferenceso that the type of power constraints is irrelevant for this issue (see also the remark at the endof this section). Unless otherwise stated, it is assumed in the remainder of this section that V ≥ is an arbitrary irreducible matrix . Due to ( ii ) of Lemma 2, this implies that the max-minSIR-balanced power vector is unique.Let us define N ( p ) := (cid:8) n ∈ N : n = arg max n ∈ N g n ( p ) = 1 (cid:9) (11)which includes the indices of those nodes for which the power constraints are active under thepower vector p . By ( i ) of Lemma 2, the cardinality of N (¯ p ) must be larger than or equal to .In what follows, let β > be the constant in part ( ii ) of the lemma. This together with part ( i ) implies that β ¯ p = ΓV ¯ p + Γz g n (¯ p ) = 1 . (12) Putting the first equation into the second one yields a set of K + 1 equations that, if written in amatrix form, show that if ¯ p solves the max-min SIR-balancing problem, then there is a constant β > such that β ˜ p = A ( n ) ˜ p , β > , ˜ p ∈ R K +1++ (13)for each n ∈ N (¯ p ) where ˜ p = (¯ p , is the extended power vector and the nonnegative matrix A ( n ) ∈ R ( K +1) × ( K +1)+ is defined to be A ( n ) = ΓV Γz P n c Tn ΓV P n c Tn Γz , n ∈ N . (14)Alternatively, we can write (12) as β ¯ p = ΓV ¯ p + Γz · g n (¯ p ) , from which we obtain, for each n ∈ N (¯ p ) , β ¯ p = B ( n ) ¯ p , β > , ¯ p ∈ R K +1++ (15)where B ( n ) ∈ R K × K + is defined to be (for each n ∈ N ) B ( n ) := ΓV + 1 P n Γzc Tn = Γ (cid:0) V + 1 P n zc Tn (cid:1) = Γ ˜ V ( n ) (16)and ˜ V ( n ) := V + P n zc Tn , n ∈ N . So, given ¯ p defined by (9), the equations (13) and (15)hold for each n ∈ N (¯ p ) . In other words, the solution of (9) in a network entirely coupled byinterference must satisfy (13) and (15) for each node n ∈ N whose power constraints are activeat the maximum. This is summarized in the following lemma. Lemma 3: If V ≥ is irreducible and ¯ p solves the max-min SIR-balancing problem (9), then ¯ p satisfies both (13) and (15) for some β > and each n ∈ N (¯ p ) .Note that the lemma is an immediate consequence of parts ( i ) and ( ii ) of Lemma 2, fromwhich (13) and (15) follow for an arbitrary n ∈ N (¯ p ) . Now we are in a position to prove thefollowing result. Lemma 4:
Suppose that V ≥ is irreducible. Then, for any constants c > and c > , thefollowing holds.(i) For each n ∈ N , there is exactly one positive vector p = p ( n ) ∈ R K + with g n ( p ) = c satisfying β ( n ) p = B ( n ) p for some β ( n ) > . Moreover, β ( n ) is a simple eigenvalue of B ( n ) and β ( n ) = ρ ( B ( n ) ) . (ii) For each n ∈ N , there is exactly one positive vector ˜ p = ˜ p ( n ) ∈ R K +1+ with ˜ p K +1 = c satisfying β ( n ) ˜ p = A ( n ) ˜ p for some β ( n ) > . Moreover, β ( n ) is a simple eigenvalue of A ( n ) and β ( n ) = ρ ( A ( n ) ) . Proof:
The reader can find the proof in Appendix VIII-C.The lemma says that, for each n ∈ N , the matrix equation β ( n ) p = B ( n ) p with β ( n ) > and p ∈ R K + is satisfied if and only if p is a positive right eigenvector of B ( n ) associated with β ( n ) = ρ ( B ( n ) ) . Furthermore, if g n ( p ) = 1 , then p is unique. Similarly, β ( n ) p = A ( n ) p with β ( n ) > and p ∈ R K +1+ is satisfied if and only if p is a positive right eigenvector of A ( n ) associated with β ( n ) = ρ ( A ( n ) ) and there is exactly one such an eigenvector whose last entryis equal to one. Furthermore, for each n ∈ N , β ( n ) = ρ ( A ( n ) ) = ρ ( B ( n ) ) . This is becauseif ρ ( B ( n ) ) p = B ( n ) p holds for some n ∈ N , then we must have ρ ( B ( n ) )˜ p = A ( n ) ˜ p where ˜ p = ( p , ∈ R K ++ . Thus, by Lemma 4, we must have ρ ( A ( n ) ) = ρ ( B ( n ) ) , n ∈ N . (17)Note that a solution to the max-min SIR-balancing problem is not necessarily obtained for each n ∈ N since, in the optimum, some power constraints may be inactive. Indeed, in general, theset N c (¯ p ) = N \ N (¯ p ) is not an empty set where ¯ p defined by (9) is unique due to ( ii ) ofLemma 2.Now we combine Lemmas 3 and 4 to obtain the following. Theorem 1:
Let β be given by (10). If V ≥ is irreducible, then the following statementsare equivalent.(i) ¯ p ∈ P + solves the max-min SIR-balancing problem (9).(ii) For each n ∈ N (¯ p ) , ¯ p is a unique positive right eigenvector of B ( n ) associated with β = ρ ( B ( n ) ) > such that g n (¯ p ) = 1 .(iii) For each n ∈ N (¯ p ) , ˜ p is a unique positive right eigenvector of A ( n ) associated with β = ρ ( A ( n ) ) > such that ˜ p K +1 = 1 . Proof:
The proof can be found in Appendix VIII-D.Theorem 1 implies that if V is irreducible, then ¯ p > is the (positive) right eigenvector of B ( n ) associated with ρ ( B ( n ) ) ∈ σ ( B ( n ) ) for each n ∈ N (¯ p ) . Alternatively, ¯ p can be obtainedfrom ˜ p = (¯ p , , which is the positive right eigenvector of A ( n ) associated with ρ ( A ( n ) ) for each n ∈ N (¯ p ) . The problem is, however, that N (¯ p ) is not known as this set is determined by the solution to the max-min SIR-balancing problem, and hence its determination is itself a part ofthe problem. As the SIR targets are feasible if and only if they are met under ¯ p , the followingcharacterization of the set N (¯ p ) immediately follows from [28] and (17). Theorem 2 ([28]):
Suppose that V ≥ is irreducible. Then, N (¯ p ) = (cid:8) n ∈ N : n = arg max n ∈ N ρ ( B ( n ) ) (cid:9) (18)Moreover, the feasible SIR region F γ is characterized as F γ = (cid:8) γ ∈ R K ++ : max n ∈ N ρ ( B ( n ) ) = max n ∈ N ρ ( A ( n ) ) ≤ , Γ = diag( γ , . . . , γ K ) (cid:9) . (19)The characterization of the feasible SIR region in (19) can be deduced directly from [28] asthe authors show that min k ∈ K (SIR k ( p ) /γ k ) ≤ min n ∈ N (1 /ρ ( B n )) , without characterizing, however, thecorresponding power allocation vector p .Theorems 1 and 2 directly lead to the following procedure for computing the max-min SIRpower vector ¯ p given by (2). Input: Γ = diag( γ , . . . , γ K ) . Output: ¯ p ∈ P + ⊂ P Find an arbitrary index n ∈ N such that n = arg max n ∈ N ρ ( B ( n ) ) where B ( n ) is given by(16). Let ¯ p be given by ρ ( B ( n ) )¯ p = B ( n ) ¯ p and normalized such that c Tn ¯ p = P n . Remark 1 (Remark on reducible matrices):
We point out that all the statements in this paperhold if B ( n ) is irreducible for each n ∈ N , which may be satisfied even if V is reducible. This isfor instance true in the case of a sum power constraint ( C = (1 , . . . , ) where B = B (1) , N = { } , is a positive matrix. Moreover, the statement of Theorem 1 can be shown to be true evenif B ( n ) is irreducible only for n ∈ N (¯ p ) .V. A PPLICATIONS
In this section, we discuss two other applications of the results. In doing so, V is assumedto be irreducible. Under this assumption, the feasible QoS region given by (7) can be shown tobe strictly convex [35, Corrolary 4.3]. A. Computation via utility-based power control
In this section, we show that ¯ p can be obtained by maximizing the following aggregate utilityfunction F ( p , w ) := X k ∈ K w k φ (cid:0) SIR k ( p ) /γ k (cid:1) (20)provided that the weight vector w = ( w , . . . , w K ) > is chosen appropriately and φ : R ++ → Q satisfies (A.2) and (A.3). To this end, define p ∗ ( w ) := arg max p ∈ P + F ( p , w ) . (21)Although P + is not compact, it can be shown [15] that the maximum exists if (A.2) and (A.3) arefulfilled. Furthermore, it is obvious that in the maximum, at least one power constraint is active,that is, Cp ∗ ≤ ˆ p holds at least with one equality. Thus, we have q ∗ ( w ) = ( q ∗ ( w ) , . . . , q ∗ K ( w )) ∈ ∂ F for q ∗ k ( w ) = φ (cid:0) SIR k ( p ∗ ( w )) /γ k (cid:1) . (22)In words, p ∗ = p ∗ ( w ) corresponds to a boundary point of F defined by (7). Different boundarypoints can be achieved by choosing different weight vectors in (20). In particular, Lemma 1implies that q ∈ ∂ F if and only if q = q ∗ ( w ) for some w > . For the analysis in this section,it is important to recall from Observation 1 that there is a bijective map from F onto P and thismap can be shown to be [15, Section 5.3] p ( q ) = ( I − G ( q ) ΓV ) − G ( q ) Γz , q ∈ F (23)where G ( q ) := diag( g ( q ) , . . . , g ( q K )) with g ( x ) defined by (A.3) and ρ ( G ( q ) ΓV ) < , whichensures the existence of p ( q ) and is satisfied for every q ∈ F [15, Section 5.3]. Note that g isstrictly increasing and g ( x ) > for all x ∈ Q , which follows from (A.2) and (A.3). Lemma 5: q ∈ ∂ F if and only if max n ∈ N λ n ( q ) = 1 where λ n ( q ) := ρ ( G ( q ) B ( n ) ) >ρ ( G ( q ) ΓV ) > . Proof:
The proof is deferred to Appendix VIII-E.Now we are in a position to prove the following.
Theorem 3:
Suppose that (A.2) and (A.3) hold. Let q ∈ ∂ F and u ( q ) = ( g ′ ( q ) /g ( q ) , . . . , g ′ ( q K ) /g ( q K )) > . Then, we have q = q ∗ ( w ) given by (22) if w = c · u ( q ) ◦ y ◦ x , c > (24) where y and x are positive left and right eigenvectors of G ( q ) B ( n ) , respectively, associatedwith λ n ( q ) for any n = arg max n ∈ N λ n ( q ) . Proof:
The reader can find the proof in Appendix VIII-F. w ρ ( G ( q ) B (1) ) = 1 F ρ ( G ( q ) B (2) ) = 1 q = φ (cid:0) SIR ( p ) /γ (cid:1) q = q ¯ q = q ∗ ( w ) ∂ F q = φ (cid:0) SIR ( p ) /γ (cid:1) Fig. 2. An illustration of Corollary 1. The figure shows an example of the feasible QoS region defined by (7) with userssubject to individual power constraints ( C = I ). The point ¯ q = q ∗ ( w ) with w = c y ◦ x , c > , corresponds to the uniquemax-min SIR-balanced power allocation. The weight vector w is normal to a hyperplane which supports the feasible QoS regionat ¯ q ∈ ∂ F . Now we can establish a connection between (2) and (21). The connection is illustrated byFigure 2.
Corollary 1:
Let y and x be positive left and right eigenvectors of B ( n ) associated with ρ ( B ( n ) ) for any n ∈ N (¯ p ) . If w = c y ◦ x , c > , then ¯ p = p ∗ ( w ) . Proof:
The proof can be found in Appendix VIII-G.
B. Saddle point characterization of the Perron roots
Finally, we point out that Theorem 1 gives rise to a saddle point characterization of ρ ( B ( n ) ) , n ∈ N (¯ p ) . Let Π K := { x ∈ R K + : k x k = 1 } and Π + K = Π K ∩ R K ++ . We define G : Π + K × P + → R as G ( w , p ) := X k ∈ K w k ψ (cid:0) γ k / SIR k ( p ) (cid:1) where ψ ( x ) := − φ (1 /x ) . A key ingredient in the proof of the saddle point characterization isthe following theorem, which can be deduced from [15, Sections 1.2.4–5]: Theorem 4:
Assume that (A.1)–(A.3) hold and V is irreducible. Let B = B ( n ) for any n ∈ N ,and let w = y ◦ x ∈ Π + K where y and x are positive left and right eigenvectors of B associatedwith ρ ( B ) . Then, for all p > , ψ ( ρ ( B )) ≤ X k ∈ K w k ψ (cid:16) ( Bp ) k p k (cid:17) . (25)Equality holds if and only if p = x > .Now we are in a position to present the saddle point characterization of the Perron root of B ( n ) , n ∈ N (¯ p ) , similar to the one for the noiseless case which can be found in [15]. Theorem 5:
Suppose that (A.2) and (A.3) hold, and V is irreducible. Then, ψ ( ρ ( B ( n ) )) = sup w ∈ Π + K inf p ∈ P + G ( w , p ) = inf p ∈ P + sup w ∈ Π + K G ( w , p ) where n = arg max n ∈ N ρ ( B ( n ) ) . Moreover, a point ( w ∗ , p ∗ ) is the saddle point of G ( w , p ) ifand only if p ∗ = ¯ p and w ∈ W , where W = n w ∈ Π + K : w = X n ∈ N c n w ( n ) , X n ∈ N c n = 1 , c n ≥ o and w ( n ) := y ( n ) ◦ x ( n ) ∈ Π + K with ρ ( B ( n ) ) x ( n ) = B ( n ) x ( n ) , ρ ( B ( n ) ) y ( n ) = ( B ( n ) ) T y ( n ) , ( y ( n ) ) T x ( n ) = 1 . In words, at the saddle point the power vector is equal to the max-min SIR-balanced power allocation, whereas the weight vector is any linear combination of the vectors w ( n ) for n ∈ N (¯ p ) .With Theorems 1 and 4 in hand, the proof is similar to that in [15, Section 1.2.4]. Theexistence of a saddle point is ensured by irreducibility of the gain matrix since then positiveleft and right eigenvectors exist. The uniqueness follows from the irreducibility property and thenormalizations.The reason why the theorem is of interest is that it provides a basis for the design of alternativepower control algorithms for saddle point problems that converge to ¯ p and may be amenable todistributed implementation. Basically, the idea of the algorithm is redolent of that of primal-dualalgorithms that converge to a saddle point of the associated Lagrange function [34]. Developmentof new algorithms is currently a subject of our ongoing work; the main idea, however, consistsin minimizing the function G ( w , p ) with respect to p , and simultaneously maximizing G ( w , p ) with respect to w . The straight-forward approach employs the gradient projection method, wherein order for the objective function to be convex in the power variable the substitution s = log( p ) is used. Each iteration encompasses the calculation of the gradient of G ( w , e s ) , and an update ofthe vectors w and s in the direction and against the direction of the gradient, respectively. Thisprocess requires a suitable step size to be chosen. The iteration is concluded with a projectionof the updated values of w and s onto the corresponding sets of valid values.The minimization of G ( w , p ) in the power domain using the gradient projection method cor-responds to employing the power control algorithm presented in [15, Section 6.5]. In particular,the gradient can be computed in a distributed way, and the projection onto the feasible set is alsodistributedly implementable in many cases of interest. As for the optimization in the weightsdomain, the gradient can be computed independently by each node, but performing the projectionrequires in general centralized operation.VI. C ONCLUSIONS
In this paper, we have characterized the max-min SIR-balanced power allocation under generalpower constraints. This characterization is an extension of the results known previously forthe noiseless case, in which the power constraints play no role. We have also established aconnection between the max-min SIR-balancing power allocation problem and the utility-basedpower allocation problem for the considered case, and as an application of our results we havediscussed two classes of power allocation algorithms based on those two approaches. Finally, wehave presented a saddle-point characterization of the Perron roots of the extended gain matriceswhich may constitute a basis for developing distributed power control algorithms.VII. A
CKNOWLEDGMENTS
We thank our colleagues Dr. Chee Wei Tan, Prof. Mung Chiang and Prof. R. Srikant forbringing to our attention their Infocom’09 paper [38], after the publication of our technicalreport [39] on which the present paper is fully based. In their paper [38], they have independentlyobtained some related results. We also refer the reader to [40] for additional views on the issues.Finally, as explained in [15], Theorem 4 of this paper is substantially related to results of theseminal paper of Friedland & Karlin [41] to which also the reader is referred. VIII. A
PPENDIX
A. Proof of Observation 2
Let (A.3) be satisfied, and let ¯ p ∈ P + be any solution to (2) or, equivalently, (6). Let ¯ q k = φ (SIR k (¯ p ) /γ k ) , k ∈ K . By Observation 1, F is a convex downward comprehensive set and,by (A.2), (6) and (7), ¯ q = (¯ q , . . . , ¯ q K ) ∈ ∂ F is its boundary point since at least one powerconstraint is active in the optimum (see Lemma 2). Thus, by irreducibility of V and Lemma1, there exists w > such that w T (¯ q − u ) ≥ for all u ∈ F . Due to positivity of w , thisimplies that for any u ∈ F , u = ¯ q , there is at least one index i = i (¯ q , u ) ∈ K such that ¯ q i > u i . In particular, for any u ∈ F , u = ¯ q ′ there is i = i (¯ q ′ , u ) ∈ K such that ¯ q ′ i > u i where ¯ q ′ k = φ (SIR k (¯ p ′ ) /γ k ) , k ∈ K . On the other hand, however, we have ¯ q ′ ≤ ¯ q . This is simplybecause ¯ q ′ = · · · = ¯ q ′ K = min k ∈ K φ (SIR k (¯ p )) . Combining both inequalities shows that ¯ q = ¯ q ′ ,and hence, by bijectivity, we obtain ¯ p = ¯ p ′ , which is unique by [15, Theorem A.51]. B. Proof of Lemma 2
Part (i) should be obvious since if we had g n (¯ p ) < for all n ∈ N , then it would bepossible to increase min k ∈ K SIR k (¯ p ) /γ k by allocating the power vector c ¯ p ∈ P + with c =1 / max n ∈ N g n (¯ p ) > . In order to show part ( ii ) , note that if (A.2) and (A.3) hold, then, byObservation 1, F is a convex downward comprehensive set. Moreover, ¯ p ∈ P + given by (6)corresponds to a point ¯ q ∈ ∂ F , with ¯ q k = φ (SIR k (¯ p ) /γ k ) , k ∈ K . Thus, by irreducibility of V ,it follows from Lemma 1 that ¯ q is a maximal point of F , and hence ¯ q ≤ q for any q ∈ F impliesthat q = ¯ q [34]. That is, there is no vector in F that is larger in all components than ¯ q . Onthe other hand, by the discussion in Sect. III, ¯ q is a point where the hyperplane in the directionof the vector (1 /K, . . . , /K ) intersects the boundary of F . As a result, ¯ q = · · · = ¯ q K , whichtogether with the maximality property and strict monotonicity of φ , shows that SIR k (¯ p ) /γ k = β for each k ∈ K where β is positive due to ( i ) . If V is irreducible, the uniqueness of ¯ p followsfrom Observation 2. C. Proof of Lemma 4
Let n ∈ N be arbitrary. First we prove part ( i ) . Since /P n zc Tn ≥ and V is irreducible,we can conclude from (16) that B ( n ) ≥ is irreducible as well. Thus, by the Perron-Frobenius theorem for irreducible matrices [33], [32], there exists a positive vector p which is an eigenvectorof B ( n ) associated with ρ ( B ( n ) ) , and there are no nonnegative eigenvectors of B ( n ) associatedwith ρ ( B ( n ) ) other than p and its positive multiples. Among all the positive eigenvectors, thereis exactly one eigenvector p > such that g n ( p ) = c . This proves part ( i ) . In order to prove ( ii ) , note that if A ( n ) was irreducible, then we could invoke the Perron-Frobenius theorem andproceed essentially as in part ( i ) to conclude ( ii ) (with the uniqueness property resulting fromthe normalization of the eigenvector so that its last component is equal to c > ). In order toshow that A ( n ) is irreducible, let G ( A ( n ) ) be the associated directed graph of { , . . . , K + 1 } nodes [33]. Since ΓV is irreducible, it follows that the subgraph G ( ΓV ) is strongly connected[33]. Furthermore, as the vector Γz is positive, we can conclude from (14) that there is a directededge leading from node K + 1 to each node n < K + 1 belonging to the subgraph G ( ΓV ) .Finally, note that as ΓV is irreducible, each row of ΓV has at least one positive entry. Hence,the vector /P n c Tn ΓV has at least one positive entry as well, from which and (14) it follows thatthere is a directed edge leading from a node belonging to G ( ΓV ) to node K + 1 . So, G ( A ( n ) ) is strongly connected, and thus A ( n ) is irreducible. D. Proof of Theorem 1 ( i ) → ( ii ) : By Lemma 3, ¯ p ∈ P + satisfies (15) for some β > . Thus, by Lemma 4, part ( i ) implies part ( ii ) . ( ii ) → ( iii ) : Given any n ∈ N (¯ p ) , it follows from (15) that ρ ( B ( n ) )¯ p = B ( n ) ¯ p with g n (¯ p ) = 1 is equivalent to ρ ( B ( n ) )¯ p = ΓV ¯ p + Γz , which in turn can be rewritten to give(13) with β = ρ ( B ( n ) ) and ˜ p = (¯ p , . Since ¯ p is positive, so is also ˜ p . Thus, ˜ p with ˜ p K +1 = 1 is a positive right eigenvector of A ( n ) and the associated eigenvalue is equal to ρ ( B ( n ) ) > . So,considering part ( iii ) of Lemma 4, we can conclude that ( iii ) follows from ( ii ) . ( iii ) → ( i ) : ByLemma 4, for each n ∈ N (¯ p ) , there exists exactly one positive vector ˜ p with ˜ p K +1 = 1 suchthat (13) is satisfied. Furthermore, β = ρ ( A ( n ) ) , n ∈ N (¯ p ) is a simple eigenvalue of A ( n ) . Nowconsidering Lemma 3 proves the last missing implication. E. Proof of Lemma 5
By (7) with (A.2), we have q ∈ F if and only if there is p ∈ P such that φ (SIR k ( p ) /γ k ) ≥ q k for each k ∈ K . Thus, q ∈ F if and only if /λ := max p ∈ P min k ∈ K (SIR k ( p ) /γ k g ( q k )) ≥ where the maximum always exists. Comparing the left hand side of the inequality above with (2) shows that the only difference to the original problem formulation is that γ k is substituted by γ k g ( q k ) or, equivalently, Γ by G ( q ) Γ , which is positive definite as well. Thus, by (10), Theorem1 and Theorem 2, we have q ∈ F if and only if λ = max n ∈ N λ n ( q ) ≤ . Moreover, p ( q ) givenby (23) is the unique power vector such that q k = φ (SIR k ( p ( q )) /γ k ) for each k ∈ K . Since theNeumann series converges for any q ∈ F , we have p ( q ) = P ∞ l =0 ( G ( q ) ΓV ) l G ( q ) Γz . Now as G ( q ) Γz is positive and G ( q ) ΓV is irreducible, we can conclude from [15, Lemma A.28] and(A.2) that each entry of p ( q ) is strictly increasing in each entry of q . Thus, as F is downwardcomprehensive and q / ∈ ∂ F holds if and only if all power constraints are inactive, for every q ∈ int(F) , there is ˜ q ∈ ∂ F such that ˜ q = q + u for some u > . By irreducibility of B ( n ) , thisimplies that λ n ( q ) < λ n ( q + u ) = λ n (˜ q ) ≤ for each n ∈ N So, if max n ∈ N λ n ( q ) = 1 , then q ∈ ∂ F . Conversely, if q ∈ ∂ F , we must have max n ∈ N λ n ( q ) = 1 since otherwise there wouldexist ˜ q / ∈ F such that max n ∈ N λ n (˜ q ) = 1 , which would contradict Theorem 2. This completesthe proof. F. Proof of Theorem 3
Let ˜ q ∈ ∂ F and n = arg max n ∈ N λ n ( q ) be arbitrary and note that F is a convex set. So, byLemma 1, there is w > such that ˜ q maximizes q w T q over F . Lemma 5 implies that thisconvex problem can be stated as ˜ q = arg max q w T q subject to λ n ( q ) = 1 , q ∈ Q K . Due to(A.2), the spectral radius is continuously differentiable on Q K . Thus, the Karush-Kuhn-Tuckerconditions [34], which are necessary and sufficient for optimality (due to the convexity property),imply that w is parallel with ∇ λ n ( q ) . Now, by [42], we have ∂λ n ( q ) ∂q k = y k g ′ ( q k ) P l ∈ K b ( n ) k,l x l = g ′ ( q k ) g ( q k ) y k P l ∈ K g ( q k ) b ( n ) k,l x l = λ n ( q ) g ′ ( q k ) g ( q k ) y k x k = g ′ ( q k ) g ( q k ) y k x k for each k ∈ K where y and x areleft and right positive eigenvectors of G ( q ) B ( n ) associated with λ n ( q ) , which, by irreducibility,are unique up to positive multiples. G. Proof of Corrolary 1 As V is irreducible, Observations 1 and 2 (see also the proof) imply that ¯ p corresponds to apoint ¯ q ∈ ∂ F . Since q ∗ ( w ) ∈ ∂ F for any w > , it follows from Theorem 3 that ¯ q = q ∗ ( w ) if w is has the form (24). Now by Observations 1 and 2, we have ¯ q = · · · = ¯ q K . Thus, as both g and its derivative g ′ are strictly monotonic (by (A.2) and (A.3)), we must have u (¯ q ) = a , a > and G (¯ q ) = 1 /ρ ( B ( n ) ) I . Thus, the corollary follows from Theorem 3. R EFERENCES [1] J. M. Aein, “Power balancing in systems employing frequency reuse,”
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