The Weighted Sum Rate Maximization in MIMO Interference Networks: Minimax Lagrangian Duality and Algorithm

Transcription

1 1 The Weighted Sum Rate Maximization in MIMO Interference Networks: Minimax Lagrangian Duaity and Agorithm Lijun Chen Seungi You Abstract We take a new approach to the weighted sumrate maximization in mutipe-input mutipe-output MIMO interference networks, by formuating an equivaent max-min probem. This reformuation has significant impications: the Lagrangian duaity of the equivaent max-min probem provides an eegant way to estabish the sum-rate duaity between an interference network and its reciproca, and more importanty, suggests a nove iterative minimax agorithm with monotonic convergence for the weighted sum-rate maximization. The design and the convergence proof of the agorithm use ony genera convex anaysis. They appy and extend to other max-min probems with simiar structure, and thus provide a genera cass of agorithms for such optimization probems. This paper presents a promising step and ends hope for estabishing a genera framework based on the minimax Lagrangian duaity for deveoping efficient resource aocation and interference management agorithms for genera MIMO interference networks. Index Terms Iterative minimax agorithm, Lagrangian duaity, max-min optimization, weighted sum-rate maximization, interference networks, mutipe-input mutipe-output MIMO. I. INTRODUCTION The weighted sum-rate maximization, which aims to maximize the weighted sum-rate of a users or data inks in a network, is an important probem that serves as a basis for many resource management and network design probems. It has a ong history, with a rich iterature from the cassica water-fiing structure for parae Gaussian channes to more recent iterative weighted MMSE agorithm [2], [15] and poite water fiing agorithm [11] for MIMO interference channes, to just name a few. The weighted sum-rate maximization is in genera a highy nonconvex and NP hard probem, and despite its importance and ong history, remains open for genera channes/networks. In this paper, we consider the weighted sum-rate maximization in a genera MIMO interference network that consists of a set of interfering data inks, each of them equipped with mutipe antennas at the transmitter and receiver. The MIMO interference network, under many different names such as MIMO B-MAC and MIMO IBC, incudes broadcast channes, mutipe access channes, interference channes, sma ce networks, and many other practica wireess networks L. Chen is with Computer Science and Teecommunications, University of Coorado, Bouder, CO 839, USA emai: ijun.chen@coorado.edu. S. You is with Contro and Dynamica Systems, Caifornia Institute of Technoogy, Pasadena, CA 91125, USA emai: syou@catech.edu. as specia cases. Specificay, we study the weighted sumrate maximization with genera inear power covariance matrix constraints, assuming Gaussian transmit signa, Gaussian noise, and the avaiabiity of channe state information at the transmitter Section II. It typifies a cass of probems that are key to the next generation wireess communication networks where the interference is a imiting factor. Various soution approaches have been proposed for the weighted sum-rate maximization in the MIMO interference network or its specia cases; see, e.g., [2] [16], [19], [21] [23], [25] [27]. Broady speaking, most of these approaches fa into the foowing three main categories, among others. The first category expoits the reation between the mutua information and the minimum mean square error MMSE, and soves the weighted sum-rate maximization based on the weighted MMSE; see, e.g., [2], [15]. The second category expoits the water-fiing structure or its variants at the optimum to sove iterativey for the KKT conditions; see, e.g., [11], [12]. The third category is based on the iterative convex approximation that at each iteration inearizes the nonconvex term around the point from the previous iteration; see, e.g., [3], [13], [21]. Many of the resuting agorithms are meta agorithms that require soving a arge convex optimization probem at each iteration, which may incur a high computationa compexity; and some of them do not even have guaranteed convergence. In this paper, we take a different approach to the weighted sum-rate maximization in the MIMO interference network, which eads to a new and efficient agorithm with guaranteed monotonic convergence as we as an eegant way to estabish the rate duaity between an interference network and its reciproca. Specificay, we reformuate the weighted sum-rate maximization as an equivaent max-min probem, by treating the interference-pus-noise covariance matrix definition as a constraint. We then construct an extended difference of ogdet function and appy matrix anaysis techniques from robust contro to estabish an expicit sadde point soution for the Lagrangian of the equivaent max-min optimization Section III-A and the Appendix. When the expicit soution is appied to the optima dua variabe, the Lagrangian duaity of the equivaent max-min probem provides an eegant way to estabish the sum-rate duaity between the interference network and its reciproca Section III. More importanty, the expicit sadde soution has significant agorithmic impication,

2 2 and suggests a nove agorithm, termed the iterative minimax agorithm, for the weighted sum-rate maximization Section IV. The design and the convergence proof of the agorithm use ony genera convex anaysis. They appy and extend to other max-min probems where the objective function is concave in the maximizing variabes and convex in the minimizing variabes and the constraints are convex, and thus provide a genera cass of agorithms for such optimization probems. The iterative minimax agorithm we design is based argey on an expicit sadde point soution for the Lagrangian of certain max-min optimization Section III-A. This expicit soution has been identified for the case where the matrices invoved are a square and invertibe in [24]. In contrast, we estabish the expicit soution for any genera matrices, as ong as the objective function is we-defined in a proper sense the Appendix. Our proof uses ony genera matrix anaysis, and the construction and techniques used are expected to find appications in handing singuarity issues that arise from the matrix form capacity formua. This paper benefits from the insight from the semina work by Yu [24] that estabishes upink-downink duaity via minimax duaity for the sum capacity of the Gaussian broadcast channe, and Section III can be seen as a substantia extension of [24]. Our mode is more genera and the resuts expect to find broad appications, and we estabish the expicit sadde point soution for the Lagrangian of the max-min optimization with genera matrices, and most importanty, we expore the agorithmic impication of the minimax duaity and expicit sadde point soution to deveop a nove agorithm for the weighted sum-rate maximization. The paper is organized as foows. The next section presents detais of the system mode and probem formuation. Section III expores the minimax Lagrangian duaity, and estabishes the expicit sadde point soution for the Lagrangian of the equivaent max-min probem and the sum-rate duaity. Section IV expores the agorithmic impication of the expicit sadde point soution, and presents the iterative minimax agorithm and its convergence anaysis. Section V provides numerica exampes to compement the theoretica anaysis in the previous sections, and Section VI concudes the paper. Notations. The capita etters such as L are used to denote sets, the capita etters in bod such as H are used to denote matrices, and the ower case etters in bod such as x are used to denote vectors. The identity matrix is denoted by I, and the zero matrix is denoted by. The trace of matrix A is denoted by Tr A. For two n n Hermitain matrices A and B, A B and A B refer to the generaized inequaities under the positive semidefinite cone S n +. The inequaity A or A then means that matrix A is positive semidefinite or positive definite, i.e., A S n + or A S n ++. II. SYSTEM MODEL Consider a genera interference network N with a set L of MIMO data inks or users, with the transmitter t and receiver r of ink L being equipped with n and m antennas respectivey. Let x C n 1 denote the transmit signa of ink, which is assumed to be circuary symmetric compex Gaussian. 1 The received signa y C m 1 at the receiver r can be written as y = k L H k x k + w, 1 where H k C m n k denotes the channe matrix from the transmitter t k to the receiver r, and w C m 1 denotes the additive circuary symmetric compex Gaussian noise with identity covariance matrix. The interference network described above is very genera and incudes as specia cases many practica channes and networks such as broadcast channes, mutipe access channes, sma ce networks, and heterogeneous networks, etc. A. The power covariance matrix constraints Denote by Σ the covariance matrix of the transmit signa x, L. We now specify the constraints on these power covariance matrices. Assume that the inks are grouped into a set S of nonempty subsets L s, s S that cover a of L. Each subset L s may correspond to those inks that are controed or managed by a certain entity or for a certain purpose. These subsets may overap with each other; and some of them may even be identica, corresponding to the situation where there may be mutipe constraints on the same subset of inks. For each ink L, denote by S the set of those subsets that incude the ink, i.e., S = {s S L s }. Each ink L is associated with an n n constraint matrix Q s for each s S ; and two of these matrices may be identica. We assume that each group of inks L s, s S is subject to a inear power covariance matrix constraint as foows: Tr Σ Q s 1, s S. 2 L s The constraint 2 is very genera and captures a reasonabe inear constraints on power. For instance, when there is ony a budget on the tota power of a inks as considered in many existing work such as [11], the cardinaity S = 1 and Q s = 1 I. When there is ony a per-ink power budget p, L, each group L s contains ony one ink and Q s = 1 P I. Each group L s, s S may aso represent those inks or users in a ce of a microce network and each ce s is subject to a tota power budget P s. In this scenario, the subsets L s are non-overapping and Q s = 1 P s I, L s. Remark 1. We have assumed inear constraints on the power covariance matrices. However, as wi be seen ater, our theory deveopment and agorithm design are based on genera convex anaysis. So the resuts in this paper can be extended to the network with noninear convex power covariance matrix constraints, which we wi investigate in future work. 1 The assumption of circuary symmetric compex signa can be dropped by appying the theory deveopment and proposed agorithm to rea Gaussian signa with twice the dimension.

3 3 B. The weighted sum-rate maximization Assume that the channe state information is avaiabe at the transmitter. For given power covariance matrix Σ, L, an achievabe rate c of the ink is given by 1 c = og I + H Σ H + I + H k Σ k H + k, 3 k L\{} where denotes the matrix determinant and the interferences from other inks are treated as noise. 2 Assume that each ink L is associated with a weight w >. We aim to aocate power for each ink so as to maximize the weighted sum-rate subject to the power covariance matrix constraints: max Σ s.t. w c 4 L L s Tr Σ Q s 1, s S. 5 The weighted sum-rate maximization is in genera a hard nonconvex probem. It is a fundamenta probem that serves as a basis for many resource management and network design probems, whie sti remains open for genera channes/networks. III. THE MINIMAX LAGRANGIAN DUALITY In this section, we wi reformuate the weighted sum-rate maximization as an equivaent max-min probem, by treating the interference-pus-noise covariance matrix definition as a constraint. This seemingy trivia reformuation has significant impications: the Lagrangian duaity of the equivaent maxmin probem provides an eegant way to estabish the sum-rate duaity between an interference network and its reciproca, and more importanty, suggests a new agorithm for the weighted sum-rate maximization. A. The minimax Lagrangian duaity Denote by Ω, L the interference-pus-noise covariance matrix at the receiver r, i.e., Ω = I + H k Σ k H + k. 6 k L\{} We can rewrite the weighted sum-rate maximization 4-5 equivaenty as the foowing max-min probem: w og Ω + H Σ H + Σ Ω og Ω 7 L s.t. Tr Σ Q s 1, s S, 8 L s Ω = I + H k Σ k H + k, L. 9 k L\{} Note that, when H Σ H + is not of fu rank, the above probem is equivaent to a truncated system where Ω is 2 If the interference from ink k to ink is competey canceed using successive decoding and canceation or dirty paper coding, we can simpy set H k =. This aows our mode to cover a wide range of communication techniques. restricted to Ω = H X H +, X. Intuitivey, this foows from the fact that when the signa at a channe is zero, it does not matter what the interference-pus-noise is, in terms of the achieved rate; mathematicay, this causes technica difficuty regarding singuar matrices; see the Appendix for more detai and insight. The objective function of probem 7-9 FΣ, Ω = w og Ω + H Σ H + og Ω L is concave in Σ and convex in Ω. Consider the Lagrangian LΣ, Ω, Λ, µ = FΣ, Ω + s S µ s 1 + Tr Λ Ω I L L s Tr Σ Q s k L\{} H k Σ k H + k, where µ = {µ s } s S with µ s the dua variabe associated with the power covariance matrix constraint 8, and Λ = {Λ } L with Λ the dua variabe associated with the interference-pus-noise covariance matrix definition 9. 3 For any given Λ, µ, L is concave in Σ and convex in Ω as F is. Thus, max Σ min Ω L = min Ω max Σ L, and the optimum in soving for the dua function is a sadde point. Consider the first order condition part of the KKT condition [1] for the optimum: 4 w H + Ω + H Σ H + 1 H = Φ, 1 w Ω 1 Ω + H Σ H + 1 = Λ, 11 where Φ = s S µ s Q s + k L\{} H + k Λ kh k. For any given feasibe dua variabe Λ, µ, the above condition gives the sadde point condition of Lagrangian L as a function of Σ, Ω; and when Λ, µ is a dua optimum, soving the equations 1-11 gives a prima optimum [1]. In the next section, we wi expoit this fact to design a nove agorithm to sove the weighted sum-rate maximization. Theorem 1. Given feasibe dua variabes Φ, µ, an expicit soution Σ, Ω for the sadde point equations 1-11 is given by: Ω = w H Φ + H + Λ 1 H H +, 12 Σ = w Φ 1 Φ + H + Λ 1 H. 13 The soution is motivated by [24] which focuses on a prima-dua optimum and where correspondingy the 3 Even though the equation 9 is an equaity constraint, the dua feasibiity requires Λ, as min Ω L = otherwise. 4 Note that the first oder condition does not hod for a dua variabes, but ony for those that satisfy the dua feasibiity condition. We ony need to consider those feasibe dua variabes [1].

4 4 optima power covariance matrix Σ and the interference-pusnoise matrix Ω are assumed to be positive definite and the channe matrix H is assumed to be square and invertibe. Here, the expicit soution is estabished for any given feasibe dua variabes, and the power covariance matrix and the interference-pus-noise matrix are positive semidefinite and the channe matrix can be any genera matrix. However, the soution is for an equivaent, truncated system where we ignore the interference-pus-noise of a channe whose signa is zero, and -1 denotes pseudo inverse if the matrix invoved is singuar. The proof of Theorem 1 is rather invoved, and is presented in the Appendix. The equations 1-11 and equations have simiar structures, which can be expoited to estabish the sumrate duaity between an interference network and its reciproca based on the Lagrangian dua of the truncated max-min probem 7-9. Definition 1. Consider an interference network N with a set L of MIMO inks and channe matrix H k from the transmitter of ink L to the receiver of ink k L. Its reciproca ˆN is defined as a network with the same set L of inks but with reversed directions where the channe matrix Ĥk from the transmitter of ink to the receiver of ink k is given by Ĥ k = H + k. The transmitter receiver of a ink in the reciproca network ˆN is the receiver transmitter of the corresponding ink in the origina network N. For instance, for a broadcast channe with a channe matrix H, its reciproca is a mutipe access channe with channe matrix H +, and vice versa [17], [18], [24]. Motivated by the structura parae between the equations 1-11 and equations 12-13, define a weighted sum-rate maximization probem for the reciproca network: ˆΣ ˆΩ s.t. w og ˆΩ + Ĥ ˆΣ Ĥ + og ˆΩ 14 L 1 Tr ˆΣ 1, s S, 15 S µ s L s ˆΩ = µ s Q s + Ĥ k ˆΣk Ĥ + k, L, 16 s S k L\{} where µ s, s S are the optima duas associated with the constraints 8, the noise covariance matrix at ink L is given by s S µ s Q s, and the power covariance matrices ˆΣ are constrained group-wise as in the origina network. Denote by ˆµ s, s S and ˆΛ, L the dua variabes associated with the constraints 15 and 16, respectivey. By Theorem 1, the foowing prima-dua of the reciproca network { ˆΣ, ˆΩ ; ˆµ s, ˆΛ } = {Λ, Φ ; µ s, Σ } 17 satisfies the first-order condition for the optimum of the weighted sum-rate maximization of the reciproca network. Even though the above probem achieves the same maxima sum-rate as probem 7-9 of the origina network, its constraints depend on the optimum of probem 7-9 and aso have a very different structure from those of probem 7-9, which makes the rate duaity between the two networks ess appeaing. In the next subsection, we wi study a few appeaing cases with strong rate duaity where the weighted sum-rate maximization probem of the reciproca network has exacty the same structure as that of the origina network. However, this weighted sum-rate maximization probem may invove the optima dua variabes of the weighted sumrate maximization of the origina network, as foows: B. Case studies We now discuss two typica cases, and show how the minimax Lagrangian duaity can be used to estabish the strong rate duaity between the interference network and its reciproca. 1 The network with the per-ink power constraints and without interink interference: Here the set S = L, Ω = I, and Q = I P with P the power budget at each ink L. As each ink is independent, we can just focus on one ink: Σ Ω s.t. og Ω + H Σ H + og Ω 18 Σ Tr 1, Ω = I. 19 The first order condition 1-11 reduces to w H + Ω + H Σ H + 1 I H = µ, P w Ω 1 Ω + H Σ H + 1 = Λ, P where µ is the dua variabe associated with the power covariance matrix constraint. Define ˆΣ = P µ Λ, ˆΩ = I. The first order condition becomes w H + Ω + H Σ H + 1 H = µ ˆΩ, 2 P w Ω 1 Ω + H Σ H + 1 = µ ˆΣ, 21 P and the expicit soution becomes w H ˆΩ + H + ˆΣ 1 H H + = µ Ω, 22 P w ˆΩ 1 ˆΩ + H + ˆΣ 1 H = µ Σ. 23 P Comparing equations 2-21 and equations 22-23, we can concude that the Lagrangian dua of the max-min probem is aso a max-min probem: ˆΣ ˆΩ s.t. og ˆΩ + H + ˆΣ H og ˆΩ 24 ˆΣ Tr 1, ˆΩ = I, 25 P which is the sum-rate maximization probem defined on the reciproca ink with channe matrix H +. At the corresponding

5 5 sadde points, the two probems achieve the same rate, since one is the dua of the other. Furthermore, introducing the dua variabes ˆµ and ˆΛ for the probem 24-25, we have the foowing correspondence: Σ ; Λ, µ = P ˆµ ˆΛ ; ˆµ P ˆΣ, ˆµ, 26 ˆΣ ; ˆΛ, ˆµ = P Λ ; µ Σ, µ. 27 µ P This recovers the we-known resut in [17], [18], [24]. The difference from [24] is that we estabish the expicit soution and the correspondence for genera power covariance matrices and channe matrices and at any sadde points of the Lagrangian function instead of ony at an optimum. 2 The network with the tota power constraint: Here S = 1 and Q = I, with the tota power budget. The max-min probem 7-9 reduces to Σ Ω s.t. w og Ω + H Σ H + og Ω 28 Tr Σ Ω = I + 1, 29 k L\{} H k Σ k H + k, 3 and the first order condition 1-11 reduces to w H + Ω + H Σ H + 1 H = Φ, w Ω 1 Ω + H Σ H + 1 = Λ, with Φ = µ I + k L\{} H+ k Λ kh k, where µ is the dua variabe associated with the tota power constraint. Define ˆΣ = µ Λ, ˆΩ = I + k L\{} H + k ˆΣ k H k. The first order condition becomes w H + Ω + H Σ H + 1 H = µ ˆΩ, 31 w Ω 1 Ω + H Σ H + 1 = µ ˆΣ, 32 and the expicit soution becomes w H ˆΩ + H + ˆΣ 1 H H + = µ Ω, 33 w ˆΩ 1 ˆΩ + H + ˆΣ 1 H = µ Σ. 34 Comparing equations and equations 33-34, we can concude that the Lagrangian dua of the max-min probem 28-3 is aso a max-min probem: w og ˆΩ + H + ˆΣ H og ˆΩ 35 ˆΣ ˆΩ ˆΣ s.t. Tr 1, 36 ˆΩ = I + k L\{} H + k ˆΣ k H k, 37 which is the weighted sum-rate maximization probem defined on a network of reciproca channes with channe matrix H +. At the corresponding sadde points, the two probems achieve the same weighted sum-rate, since one is the dua of the other. Furthermore, introducing the dua variabes ˆµ and ˆΛ for the probem 35-37, we have the foowing correspondence: Σ ; Λ, µ = ˆµ ˆΛ ; ˆΣ ; ˆΛ, ˆµ = µ Λ ; ˆµ ˆΣ, ˆµ, 38 µ Σ, µ. 39 This provides a simpe proof of the weighted sum-rate duaity for the MIMO interference network with the tota power constraint identified in, e.g., [11]. IV. THE ITERATIVE MINIMAX ALGORITHM Motivated by the minimax Lagrangian duaity, in this section we wi design a nove agorithm for the weighted sumrate maximization and estabish its convergence properties. A. The iterative minimax agorithm Note that an optimum of the max-min probem 7-9 is a sadde point, and the first order condition 1-11 and its expicit soution or part of them wi give a sadde point, maximum, or minimum of Lagrangian L when certain subset of its variabes is fixed and given. This motivates an iterative minimax agorithm to achieve an optimum, as foows. 1 Start with given Σ n, L that is feasibe, i.e., Tr Σ n Q s 1, s S, L s and Ω n = I + k L\{} H kσ n k H+ k, L. By equation 11 that gives the condition for minimizing L over Ω, we choose Λ n such that Λ n = w Ω n 1 Ω n + H Σ n H + Therefore, for any Ω, we have FΣ n, Ω n LΣ n, Ω n, Λ n, µ n 1. 4 LΣ n, Ω, Λ n, µ n, 41 where µ n wi be determined ater. Define Φ n = µ n s Q s + H + k Λn kh k. 42 s S k L\{} Note that Φ n does not necessary satisfy equation 1. 2 Given the above Λ n, Φn and µn, by equations 12-13, we choose Σ n+1, Ω n+1 such that 1 Σ n+1 = w Φ n 1 Φ n + H Λ n H + Ω n+1 = w H Φ n + H Λ n H + Pug Ω = Ω n+1 into inequaity 41, we have H FΣ n, Ω n LΣ n, Ω n+1, Λ n, µ n. 45

6 6 By the first order condition 1-11, Σ n+1, Ω n+1 is the sadde point of LΣ, Ω, Λ n, µ n. Thus, LΣ n, Ω n+1, Λ n, µ n L Σ n+1, Ω n+1, Λ n, µ n for any Ω. 3 The matrix expicity by that T = {s S Σ n+1 Σ n+1 L s Tr L Σ n+1, Ω, Λ n, µ n 46 is a function of µ n s, s S, denoted µ n s ; s S. Define the set T such Q s n+1 Σ µ n s = + ; s S 1}. For each s S\T, we set µ n s =. For those s T, we choose µ n s such that Q s n+1 Σ µ n s ; s S = 1, s T. 47 L s Tr Note that Tr Σn+1 µ n s ; s S is decreasing in µ n s, and there are T equations for T variabes. So, there exists a soution to equation 47. With the afore choice of µ n s, s S, we can see that µ n s 1 Σn+1 Q s =. 48 L s Tr The above is a compementary sackness condition part of the KKT condition that is required at an optimum [1], but in our agorithm we enforce this condition at each iteration. 4 Let λ = max s S L s Tr Σn+1 Q s. We see that < λ 1. We then choose Σ n+1, Ω n+1 such that Ω n+1 = I + k L\{} Σ n+1 = Σ n+1 λ, 49 H k Σ n+1 k H + k. 5 5 Repeat 1-4, we wi obtain a monotone increasing sequence {FΣ n, Ω n }, based on which we can concude that Σ n, Ω n converges to a sadde point of the maxmin probem 7-9 and thus an oca optimum of the weighted sum-rate maximization 4-5. We ca the above agorithm the iterative minimax agorithm; see Tabe I for a forma description. Different from many existing agorithms mentioned in Section I, the iterative minimax agorithm is not a meta agorithm that requires soving a arge convex optimization probem at each iteration. The ony step of the agorithm that may potentiay be compicated is Step 8 that may require soving sets of couped equations when the subsets L s, s S overap. But Step 8 can be soved efficienty using, e.g., the bisection search method. Moreover, in practice, we expect that the subsets L s, s S sedom overap. TABLE I THE ITERATIVE MINIMAX ALGORITHM 1 Initiaize Σ, L such that L s Tr Σ Q s 1, s S 2 Ω I + k L\{} H kσ k H + k, L 3 Λ w Ω 1 Ω + H Σ H + 1, L 4 Φ s S µsqs + k L\{} H+ k Λ kh k, L 5 Σ 1 w Φ 1 Φ + H Λ H +, L 6 T {s S L Q s Tr s Σ µ s = + ; s S 1} 7 µ s if s S\T 8 For s T, choose µ s such that L s Tr Q s Σ µ s; s S = 1, s T 9 λ max s S L s Tr Σ Q s 1 Σ Σ λ, L 11 Go to 2 Pug Σ n+1 = λσ n+1 and Ω = λω n+1 into the inequaity 46 and combine with the inequaity 45, we have FΣ n, Ω n LλΣ n+1, λω n+1, Λ n, µ n = FΣ n+1, Ω n+1 + Lλ 1Tr Λ n + s S µ n s 1 L s Tr Σn+1 Q s = FΣ n+1, Ω n+1 + Lλ 1Tr Λ n FΣ n+1, Ω n+1, 51 where the second equaity foows from equation 48 and the ast inequaity foows from the fact that λ 1. B. The convergence anaysis We now study the convergence properties of the iterative minimax agorithm. The foowing resut is immediate. Theorem 2. The iterative minimax agorithm converges to a sadde point Σ, Ω of the max-min probem 7-9; and Σ is a oca optimum of the weighted sum-rate maximization 4-5. Proof. By inequaity 51, we have With Lemma??, to show the convergence of the iterative minimax agorithm, it is enough to show that if FΣ n, Ω n = FΣ n+1, Ω n+1, then Σ n, Ω n = Σ n+1, Ω n+1. Seen form the derivation of the inequaity 51, if

7 7 FΣ n, Ω n = FΣ n+1, Ω n+1, then FΣ n, Ω n = LΣ n, Ω n, Λ n, µ n = LΣ n, Ω n+1, Λ n, µ n = L Σ n+1, Ω n+1, Λ n, µ n = LΣ n+1, Ω n+1, Λ n, µ n = FΣ n+1, Ω n+1. It foows that both Σ n, Ω n, Λ n, µ n and Σ n+1, Ω n+1, Λ n, µ n satisfy the KKT condition the first order condition, the prima feasibiity, the dua feasibiity, and the compementary sackness [1] of the max-min probem 7-9, and thus both are sadde points of the max-min probem. Furthermore, for any given dua variabes, the Lagrangian L is stricty concave in Σ. So, Σ n = Σ n+1, and Ω n = Ω n+1 foows. Therfore, the iterative minimax agorithm converges monotonicay to a sadde point of the max-min probem 7-9. The second part of the theorem foows from the equivaence between the max-min probem and the weighted sum-rate maximization probem. Remark 2. The iterative minimax agorithm converges fairy fast and can be impemented reatime. As each ink knows its own power covariance matrix and can measure/estimate its interference-pus-noise covariance matrix, the agorithm admits a distributed impementation if used as a reatime agorithm. Remark 3. The design and the convergence proof of the iterative minimax agorithm use ony genera convex anaysis. They appy and extend to any max-min probems where the objective function is concave in the maximizing variabes and convex in the minimizing variabes and the constraints are convex, and thus provide a genera cass of agorithms for such optimization probems, which we wi investigate in detai in future work. C. Case studies I We now discuss two typica cases and the corresponding iterative minimax agorithms. 1 The network with the tota power constraint: As mentioned in Section III-B2, here S = 1, and Q = with the tota power budget. The matrix Σ defined in Section IV-A is a function of µ, the dua variabe associated with the tota power constraint. The iterative minimax agorithm reduces to that described in Tabe II. We have proposed another agorithm for the network with the tota power constraint in a previous work [1], which uses the fact that the tota power constraint is tight at an optimum, and normaizes µ such that L Tr µ P Σ T = 1, i.e., the agorithm enforces the tightness of the tota power constraint at the initia point and each iteration. In contrast, the agorithm in Tabe II enforces the compementary sackness condition at each iteration and can start with any feasibe Σ. Moreover, the agorithm in [1] hardy offers any insight on the agorithm TABLE II THE ITERATIVE MINIMAX ALGORITHM FOR THE NETWORK WITH THE TOTAL POWER CONSTRAINT 1Initiaize Σ, L such that L Tr Σ 1 2Ω I + k L\{} H kσ k H + k, L Λ w Ω 1 Ω + H Σ H +, L 4Φ µi + k L\{} H+ k Λ kh k, L 5 Σ 1 w Φ 1 Φ + H Λ H +, L 6µ if L Tr Σ + < 1; otherwise, choose µ such that L Tr Σ µ = 1 7λ L Tr Σ 8Σ Σ λ, L 9Go to 2 design for the network with genera inear power covariance matrix constraints. 2 The network with the per-ink power constraints: Here the set S = L, and Q = I P with P the power budget at each ink L. The matrix Σ defined in Section IV-A is a function of µ, the dua variabe associated with the power constraint at ink. The iterative minimax agorithm reduces to that described in Tabe III. TABLE III THE ITERATIVE MINIMAX ALGORITHM FOR THE NETWORK WITH THE PER-LINK POWER CONSTRAINTS 1 Initiaize Σ, L such that Tr Σ 1 P 2 Ω I + k L\{} H kσ k H + k, L 3 Λ w Ω 1 5 Σ w Φ 1 Ω + H Σ H + 4 Φ µi + k L\{} H+ k Λ kh k, L Φ + H Λ H + 6 µ if Tr Σ + P < 1; otherwise, choose µ such that Tr 7 λ max L Tr Σ 8 Σ Σ λ, L 9 Go to 2 V. NUMERICAL EXAMPLES 1, L 1, L Σ µ P = 1 In this section, we provide numerica exampes to compement the anaysis in the previous sections. Consider a network where each ink is equipped with 3 4 antennas at its transmitter receiver and the channe matrices have zero-mean, unit-variance, i.i.d. compex Gaussian entries. We wi consider and compare the networks with ow, moderate, and high interference, which are characterized by scaing the interference channe matrices H ij, i j with a factor of.1,

8 8 1, and 5 respectivey. The weights w s are uniformy drawn from [.5, 1]. The impementation of our iterative minimax agorithm is straightforward. It ony uses basic matrix operations, except for finding µ for which we use a bisection search method. This is different from many other agorithms that need additiona probem parser or use the interior point method which are often hard to impement in practica appications. A. The network with the tota power constraint Figures 1, 2 and 3 show the monotonic convergence of our agorithm Tabe II in a network with L = 1 interfering inks and a tota power constraint = 1. Overa, the agorithm shows very fast convergence. We see that the convergence speed depends on the strength of interference. As the interference becomes stronger, the weighted sum-rate becomes highy non-convex. This intrinsic difficuty of the probem makes the convergence sower. The weighted sum rate nats Fig. 1. The network with ow interference and tota power constraint. The weighted sum rate nats Fig. 2. The network with moderate interference and tota power constraint. B. The network with the per-ink power constraints Figures 4, 5 and 6 show the monotonic convergence of our agorithm Tabe III in a network with L = 1 interfering The weighted sum rate nats Fig. 3. The network with high interference and tota power constraint. inks and per-ink power constraints where P s are uniformy drawn from {1, 2,, 1}. Again, we see that the stronger the interference, the sower the agorithm converges; but overa, the agorithm shows fast convergence. The weighted sum rate nats Fig. 4. The network with ow interference and per-ink power constraints. The weighted sum rate nats Fig. 5. The network with moderate interference and per-ink power constraints.

9 9 12 The weighted sum rate nats The weighted sum rate nats Fig. 6. The network with high interference and per-ink power constraints. C. The network with the genera inear power constraints We have tested and evauated the genera iterative minimax agorithm Tabe I for arge networks with genera inear power covariance constraints. Here we present resuts on a network with L = 3 interfering inks, i.e., the network serves the 3 inks or users simutaneousy, without using any time, frequency, or code mutipexing. These inks are partitioned into five groups: S = {{1}, {2}, {3,, 7}, {8,, 2}, {21,, 3}}. We can view these different groups of inks as being served by different ces. We consider cases with different combinations of intrace interference i.e., interference between the inks within a ce and interce interference i.e., interference between the inks of different ces. The power constraint matrix Q s is set to the scaed identity matrix without oss of generaity, and the power budget of each ce is randomy drawn from {1,, 1} and then post-mutipied by the cardinaity of the ce. The weighted sum rate nats Fig. 7. The network with ow intrace interference and ow interce interference. Figures 7 9 show the monotonic convergence of our agorithm for the cases where the intrace interference interce interference are ow ow, moderate ow, and high Fig. 8. The network with moderate intrace interference and ow interce interference. The weighted sum rate nats Fig. 9. The network with high intrace interference and moderate interce interference. moderate, respectivey. As expected, the agorithm shows fast convergence. Note that in our numerica experiments, we draw parameters such as the power budgets for the network randomy for each run of the simuation. So, the sum-rate achieved for specific exampes may not appear proportiona to the network size. For exampe, compared to Figures 5 6 of the 1-ink network, Figures 8 9 of the 3-ink network show a much ower rate per-ink, which is because exampes shown in Figures 8 9 happen to have much ower aggregate power. D. Compexity anaysis We have evauated in the above the monotonic convergence of the iterative minimax agorithm in terms of the number of iterations. We now anayze the compexity of each iteration. Reca that L is the number of data inks, and for simpicity, assume that each ink has N transmit and receive antennas, so the resuting Σ is an N N matrix. Suppose that we use the straightforward matrix mutipication and inversion, then the compexity of these operations are ON 3. In each iteration, Ω incurs a compexity of OLN 3, and so does Ω + H, Σ n+1 H +,. Furthermore, Φ incurs a compexity

10 1 of OLN 3, and so do Σ and Σ. Since we need L of these operations, the tota compexity is OL 2 N 3. If we use faster matrix mutipication such as the one in [2] that has a compexity of ON , we can reduce computationa compexity at each iteration to OL 2 N VI. CONCLUSION We have taken a new approach to the weighted sumrate maximization in the MIMO interference networks, by formuating an equivaent max-min probem. The Lagrangian duaity of the equivaent max-min probem provides an eegant way to estabish the sum-rate duaity between an interference network and its reciproca, and more importanty, suggests a nove iterative minimax agorithm with monotonic convergence for the weighted sum-rate maximization. The design and the convergence proof of the iterative minimax agorithm use ony genera convex anaysis and matrix anaysis. They appy and extend to any max-min probems where the objective function is concave in the maximizing variabes and convex in the minimizing variabes and the constraints are convex, and thus provide a genera cass of agorithms for such optimization probems. This paper presents a promising step and ends hope for estabishing a genera framework based on the minimax Lagrangian duaity for characterizing the weighted sum-rate and deveoping efficient resource aocation and interference management agorithms for genera MIMO interference networks. As further work, we wi study the practica and distributed impementation of the iterative minimax agorithm and evauate its performance under reaistic characteristics of wireess channes. We are aso studying to expoit the specia structure i.e., the objective function is convex in minimizing variabes and concave in maximizing variabes of the probem to design agorithm that is guaranteed to converge to a goba optimum. We wi aso investigate the appication of the minimax duaity and agorithm to other resource management and design probems in wireess networks. Lasty, we are exporing the agorithmic impication of the minimax duaity for genera nonconvex optimization probems that can be re-formuated equivaenty as a max-min probem. APPENDIX: PROOF OF THEOREM 1 Before we present the proof, we first define an extended difference of ogdet function. Let A, B S n +, the difference of ogdet function F A, B = og A + B og B is not we-defined if B is not positive definite. If there exists a nonsinguar square matrix T such that [ ] [ ] T + A1 AT =, T + B1 BT = where A 1 S m +, B 1 S m ++ for some m n, then we can define an extended difference of ogdet function: F A, B := og A 1 + B 1 og B 1. With the definition of the above extended function, matrix inverse resuting from the derivative of ogdet function is pseudo inverse when the matrix invoved is singuar. In the foowing, a difference of ogdet function is meant to be the extended difference of ogdet function, and matrix inverse is pseudo inverse when the matrix invoved is singuar. We now come to the proof of Theorem 1. For simpicity of presentation and without oss of generaity, we reoad notations and consider the foowing probem: Σ Ω og Ω + HΣH + og Ω + Tr ΛΩ Tr ΦΣ 52 where Λ and Φ. The key idea of the proof is to show that probem 52 is equivaent to a probem with Ω restricted to Ω = HXH +, X. Lemma 1. The probem 52 is equivaent to the foowing probem: Σ Ω og Ω + HΣH + og Ω + Tr ΛΩ Tr ΦΣ 53 s.t. Ω = HXH +, X. 54 Proof. Since HΣH + and Λ, there exists a nonsinguar square matrix T such that TΛT + = S 1 S 3, T + 1 HΣH + T 1 = S 1 S 2, where S 1, S 2, S 3 are diagona and positive definite; see, e.g., Theorem 3.22 in [28]. Let Ω = T + ΩT, probem 52 becomes Σ Ω og Ω + T + 1 HΣH + T 1 og Ω +Tr TΛT + Ω Tr ΦΣ. Now, consider those terms in the objective function that depend on Ω: L Ω = og Ω + T + 1 HΣH + T 1 og Ω + Tr TΛT + Ω Ω 11 + S 1 Ω12 Ω13 Ω14 = og Ω 22 + S 2 Ω23 Ω24 Ω + 13 Ω + 23 Ω 33 Ω34 Ω + 14 Ω + 24 Ω + 34 Ω 44 Ω 11 Ω12 Ω13 Ω14 og Ω 22 Ω23 Ω24 Ω + 13 Ω + 23 Ω 33 Ω34 Ω + 14 Ω + 24 Ω + 34 Ω 44 +Tr S 1 Ω11 + Tr S 3 Ω33

11 11 and its minimization over Ω. By the determinant [ ] formua for bock matrix, when A is invertibe A B = C D A D CA 1 B, and the fact that the determinant is a continuous function, we have L Ω Ω 11 + S 1 Ω12 Ω13 og Ω 22 + S 2 Ω23 Ω + 13 Ω + 23 Ω 33 Ω 11 Ω12 Ω13 og Ω 22 Ω23 Ω + 13 Ω + 23 Ω 33 +Tr S 1 Ω11 + Tr S 3 Ω33, where the equaity can be achieved when Ω i4 = for a i = 1, 2, 3, 4. 5 We wi restrict Ω to those with Ω i4 = for a i = 1, 2, 3, 4, as the equaity is achieved at one of those matrices. Since S 3 and Ω 33, Tr S 3 Ω33. We further have [ ] L Ω og Ω11 + S 1 Ω12 Ω 22 + S 2 [ ] og Ω11 Ω12 + Tr S 1 Ω11, Ω 22 where the equaity is achieved when additionay Ω i3 = for a i = 1, 2, 3. Therefore, we concude that there exists a minimizer Ω with the form: Ω = Ω 11 Ω12 Ω 22. The above manipuation is to restrict the probem to an equivaent, truncated system where we ignore the interferencepus-noise of a channe whose signa is zero. As mentioned in Section III-A, intuitivey, the equivaence of this truncated system to the origina max-min probem foows from the fact that when the signa is zero it does not matter what the interference-pus-noise is. Now, consider a vector v such that H + v =, we have S 1 v + HΣH T v = v + T + S 2 which impies =, Tv = [ v 3 v 4 ] T. Tv 5 By the determinant formua and the Syvester s criterion for positive semidefinite matrix, if Ω 44 =, then Ω i4 = for a i = 1, 2, 3. Therefore, v + Ω v = v + T + Ω Tv = v + T + =. Ω 11 Ω12 Ω 22 Tv This impies the nu space N H + N Ω, and further, the range RH RΩ + = RΩ. Therefore, there exists a matrix X such that Ω = HXH +. We concude that there exists an optima soution with Ω = HXH +, and thus probem 52 and probem are equivaent. With Lemma 1, we are ready to present the expicit sadde point soution. Consider the ogdet terms in the objective function: og Ω + HΣH + og Ω = og HXH + + HΣH + og HXH + = og H + HX + Σ og H + HX = og X + Σ og X. The singuarity issue comes out when H + H is not invertibe, but this can be handed by adding a sma term κi, κ > to H + H and then taking the imit κ. Thus, we can transform probem 52 into the foowing simpe one: og X+Σ og X +Tr H + ΛHX Tr ΦΣ. Σ X 55 By the first order optimaity condition for the sadde point, we have X + Σ 1 Φ =, X + Σ 1 X 1 + H + ΛH =, from which we obtain the foowing expicit sadde point soution: Σ = Φ 1 Φ + H + ΛH 1, 56 X = Φ + H + ΛH 1, and in terms of Ω we have Ω = HXH + = HΦ + H + ΛH 1 H Note that probem 55 is we-defined ony when Σ, X satisfy the property specified for matrices A, B in the beginning of this Appendix. This is verified as foows. Proposition 1. The objective function in probem 55 is wedefined for matrices Σ, X that are given by Proof. Let Ψ = Φ + H + ΛH, we have that the nu space N Ψ N Φ. To see this, note that Φ and H + ΛH. Suppose v N Ψ, then v T Ψv = v T Φv + v T H + ΛHv =. As each term is nonnegative, v T Φv =, i.e., v N Φ.

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[26] L. Zhang, Y.-C. Liang, Y. Xin, R. Zhang, and H.V. Poor. On Gaussian MIMO BC-MAC Duaity with Mutipe Transmit Covariance Constraints. In Proceedings of IEEE Internationa Symposium on Information Theory ISIT, June 29. [27] L. Zhang, Y. Xin, and Y.-C. Liang. Weighted Sum Rate Optimization for Cognitive Radio MIMO Broadcast Channes. IEEE Transactions on Wireess Communications, 86: , June 29. [28] K. Zhou, J. C. Doye, and K. Gover. Robust and Optima Contro. Prentice Ha, New Jersey, PLACE PHOTO HERE Lijun Chen M 5 is an Assistant Professor of Computer Science and Teecommunications at University of Coorado at Bouder. He received the Ph.D. from Caifornia Institute of Technoogy. He was a co-recipient of the Best Paper Award at the IEEE Internationa Conference on Mobie Ad-hoc and Sensor Systems MASS in 27. His current research interests incude optimization and contro of networked systems, distributed optimization and contro, convex reaxation and parsimonious soutions, and game theory and its engineering appication. He is currenty an editor of IEEE Transactions on Communications. PLACE PHOTO HERE Seungi You SM 12 received the B.S. degree in eectrica engineering from the Seou Nationa University, Seou, Korea, in 211. He is currenty a Ph.D. student at the Department of Computing and Mathematica Science, Caifornia Institute of Technoogy. His research interests incude information theory, contro theory, statistica signa processing, smart grids and optimization.