GRAB-N-PULL: AN OPTIMIZATION FRAMEWORK FOR FAIRNESS-ACHIEVING NETWORKS

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1 GRAB-N-PULL: AN OPTIMIZATION FRAMEWORK FOR FAIRNESS-ACHIEVING NETWORKS Mojtaba Soltanalian +, Ahmad Gharanjik, M. R. Bhavani Shankar and Björn Ottersten + Dept. of Electrical and Computer Engineering, University of Illinois at Chicago Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg ABSTRACT In this paper, e present an optimization frameork for designing precoding (a.k.a. beamforming) signals that are instrumental in achieving a fair user performance through the netorks. The precoding design problem in such scenarios can typically be formulated as a non-convex max-min fractional quadratic program. Using a penalized version of the original design problem, e derive a simplified quadratic reformulation of the problem in terms of the signal (to be designed). Each iteration of the proposed design frameork consists of a combination of poer method-like iterations and the Gram-Schmidt process, and as a result, enjoys a lo computational cost. Moreover, the suggested approach can handle various types of signal constraints such as total-poer, per-antenna poer, unimodularity, or discrete-phase requirements an advantage hich is not shared by other existing approaches in the literature. Index Terms Beamforming, fractional programming, maxmin fairness, non-convex optimization, precoding 1. INTRODUCTION A idely used proactive interpretation of fairness in ireless netorks, referred to as the max-min fairness [1 8], is to allocate the available resources in order to maximize the minimal user performance in the netork. In such scenarios, a judicious design of the precoding signals for different users can be vieed as a vital part of the netork configuration Problem Formulation We consider the precoding problem for a donlink channel, ith an M-antenna transmitter and K single-antenna users. Let h i C M denote the channel beteen the transmit antennas and the i th user. Also let i C M denote the precoding vector corresponding to the i th user. To form the data stream to the users, any complex symbol to be transmitted, ill be modulated by the precoding vector of the intended user. The precoding vectors are to be designed in order to enhance the netork performance. In particular, the signal-to-noiseplus-interference ratio (SINR) of the i th user is given by [9] H i R i i SINR i = ( ), i [K], (1) j [K]\i} H j Rij + σi here R i = Eh ih H i } is the covariance matrix of the i th channel, σi denotes the variance of the zero-mean additive hite Gaussian This ork as supported in part by the National Research Fund (FNR), Luxembourg under AFR grant (Reference ). Corresponding author msol@uic.edu noise (AWGN), and [K] = 1,,, K}. Note that by a specific reformulation, the SINR metric in (1) can be reritten as a fractional quadratic criterion. To see this, define the stacked precoding vector C N (ith N = KM) as vec([ 1 K]), () and observe that (1) ill increase for any increased scaling of. As a result, any finite-energy constraint on hile maximizing SINR i} ill be active, i.e. it ill be satisfied ith equality. Accordingly, suppose = P, and let A i R i Diag (e i), i [K], (3) B i R i (I K Diag (e i)) + σ i IN, i [K], (4) P here Diag (.) denotes the diagonal matrix formed by the entries of the vector argument, stands for the Kronecker product of matrices, and e i is the the i th standard basis vector in C K. No, it is not difficult to verify that SINR i = H A i, i [K], (5) in hich A i} are positive semidefinite (PSD) and B i} are positive definite (PD). Finally, the precoding design problem for maximizing the minimal user SINR performance can be formulated as P 1 : min H A i s. t. Ω, (6) here Ω denotes the feasible set of determined by the associated signal constraints. Note that P 1 may also be used to formulate the eighted SINR optimization problems; see [1, ] for details. 1.. Contributions of this Work The above precoding design problem has been studied extensively in the literature hen Ω denotes a total-poer or per-antenna poer constraint (see e.g., [1 8] and the references therein). As a result, different approaches have been proposed to solve the design problem, including those based on uplink-donlink duality [1], the Lagrangian duality [3] and quasi-convex formulations [8]. In this ork, e propose a novel optimization frameork (hich e call Grab-n-Pull) that can efficiently tackle P 1. Note that the proposed frameork subsumes the traditional total-poer or perantenna poer constraints, hile also alloing for intricate signal constraints such as unimodularity or discrete-phase requirements. Remark 1: It is orth mentioning that the problem formulation P 1 in (6) is also useful in other applications, including e.g., aveform design for radar [1, 11], and relay beamforming [1, 13]. }

2 . THE PROPOSED FRAMEWORK We begin by considering a reformulated version of P 1; namely, P : min λ i} s. t. Ω, (7) λ i = H A i, i [K]. (8) Note that (8) holds if and only if A 1 i = λ i B 1 i, or equiv- alently A 1 i = λ i B 1 i, here (.) 1 denotes the Hermitian square-root of the matrix argument [14]. In particular, the lefthand side of (8) is close to the right-hand side of (8) if and only if A 1 i is close to λ i B 1 i. Therefore, by employing the auxiliary variables λ i}, one can consider the folloing optimization problem as an alternative to P (and P 1): P 3 :,λ i } min λi} η K ( A 1 i λ i B 1 i ) s. t. Ω ; λ i, i [K]; (9) in hich η > determines the eight of the penalty-term added to the original objective of P ; and here P 3 and P coincide as η +. Note that optimizing P 3 ith respect to (. r. t.) may require reriting P 3 as a quartic objective in. To circumvent this, e continue by introducing P 4 yet another alternative objective: P 4 :,λ i },Q i } min λi} η K A 1 i λ iq i B 1 i s. t. Ω, λ i, i [K]; (1) Q H i Q i = IN, i [K]. (11) Toards understanding the equivalence of P 3 and P 4, observe that the maximizer Q i of P 4 is a unitary rotation matrix that aligns the vector B 1 i in the same direction as A 1 i, ithout changing its l -norm (i.e. B 1 i ). More precisely, at the maximizer Q i of P 4, e have that ( ) Q i B 1 B 1 i = i A 1 A 1 i. (1) i In contrast to P 3, the optimization problem P 4 can be easily reritten as a quadratic program (QP) in ; a idely studied type of program that facilitates the usage of poer method-like iterations, and thus employing different signal constraints Ω more on this later. In the folloing, e propose an efficient iterative optimization frameork based on a separate optimization of the objective of P 4 over its three partition of variables at each iteration, viz., Q i }, and λ i}, here the iterations can be started from any arbitrary initialization..1. Poer Method-Like Iterations (Optimization. r. t. ) For fixed Q i } and λ i}, one can optimize P 4. r. t. via minimizing the criterion: A 1 i λ iq i B 1 i = H R (13) here R = (A i + λ ib i) } λ i(a 1 i Q i B 1 i + B 1 i Q H i A 1 i ). Due to the fact that Ω enforces a fixed l -norm on (i.e. = P ), by defining the PD matrix ˆR µi R (ith µ > being larger than the maximum eigenvalue of R), e have that H R = H ˆR + µp in hich µp is constant. Consequently, one can minimize (or decrease monotonically) the criterion in (13) by maximizing (or increasing monotonically) the objective of the folloing optimization problem: H ˆR (14) s. t. Ω. Although (14) is NP-hard for a general signal constraint set [15,16], a monotonically increasing objective of (14) can be obtained using poer method-like iterations developed in [16], and [17]; namely, e update iteratively by solving the folloing nearest-vector problem at each iteration: min (s+1) ˆR (s) (15) (s+1) s. t. (s+1) Ω, here s denotes the internal, and () is the current value of. Note that e can continue updating until convergence in the objective of (14), or for a fixed number of steps, say S. No, e take a deeper look at various signal constraints Ω typically used in practice, as ell as their associated constrained solutions to (15): Total-poer constraint: In this case, Ω can be defined as Ω = : = P }, here P >. Then, the set of poer method-like iterations in (15) boils don to a typical poer method aiming to find the dominant eigenvector of ˆR, hoever ith an additional scaling to attain a poer of P. Per-antenna poer constraint: We consider M antennas ith assigned poer values P i}, and assume that K = N/M entries of are devoted to each antenna. As a result, e can solve (15) by considering the nearest-vector problem for sub-vectors associated ith each antenna separately i.e., M nearest-vector problems all ith vector arguments of length K. Unimodular signal design: The set of unimodular codes is defined as Ω = e jϕ : ϕ [, π) } N. The unimodular solution to (15) is simply given by (P = N): ( ( )) (s+1) = exp j arg ˆR (s). (16) Discrete-phase signal design: We define the set of discretephase signals as, Ω = e j π N } Q q : q =, 1,, Q 1 here Q denotes the phase quantization level. The discretephase solution to (15) is given by (P = N): ( ( (s+1) = exp (jµ ))) Q arg ˆR (s) (17) here µ Q(.) yields (for each entry of the vector argument) the closest element in the Q-ary alphabet. Finally, e refer the interested reader to find more details on the properties of poer method-like iterations in [16]- [18].

3 .. Rotation-Aided Fitting (Optimization. r. t. Q i }) Suppose and λ i} are fixed. As discussed earlier, the maximizer Q i of P 4 is a rotation matrix that maps B 1 i in the same direction as A 1 i. Let u i = A 1 i / A 1 i, v i = B 1 i / B 1 i, (18) and note that (1) can be reritten as u i = Q i v i for all i [K]. We define the unitary matrices Q ui and Q vi in C N N as Q ui = [û i1, û i,..., û in ] (19) Q vi = [ v i1, v i,..., v in ] () here each of the sets û ij} j and v ij} j, j [N], builds an orthonormal basis for C N, and û i1 = u i, v i1 = v i. Based on the above definitions, the folloing lemma constructs the optimal Q i (the proof of Lemma 1 is straightforard and omitted herein): Lemma 1. The maximizer Q i } of P 4 can be found as Q i = Q ui Q H v i for all i [K]. Remark : Note that Q ui } and Q vi } can be obtained via the Gram-Schmidt process, and are not generally unique since û ij} and v ij} can be chosen rather arbitrarily for j 1. This further implies that the maximizer Q i } of P 4 is also not unique, hich is in agreement ith the common understanding that rotation matrices are not necessarily unique hen the dimension gros large..3. Grab-n-Pull (Optimization. r. t. λ i}) Note that, once the optimal Q i } is used, the objectives of P 3 and P 4 can be considered interchangeably. We assume that the optimal and Q i } are obtained according to the above discussions, and are fixed. Therefore, to find λ i}, e can equivalently focus on obtaining the maximizer λ i} of P 3 via the optimization problem: Λ : λ i } min λ i} η ( A 1 i λ i B 1 i ) s. t. λ i, i [K]. (1) Definition 1. Let λ i } denote the optimal λ i} of Λ, and λ min λ i }. We let Υ to denote the set of all indices m for hich λ m takes the minimal value among all λ i }, i.e. Υ = m [K] : λ m = λ }. () Moreover, e refer to γi A 1 i / B 1 i as the shado value of λ i, for all i [K]. It is straightforard to verify from the objective of Λ that if λ i > λ, then λ i = γi. On the other hand, to obtain λ, e need to maximize the criterion: f(λ) = λ η ( A 1 k ) λ B 1 k. (3) Provided that η is large enough (see the loer bound in Theorem 1), the optimal λ of the above quadratic criterion (in λ) is given by λ = η α kβ k η β k 1 (4) in hich α k A 1 k, and β k B 1 k. It is interesting to have some insight into hat λ represents: Note that (4) can be reritten as λ = γ kβk. (5) β k 1/η As a result, λ can be vieed as a eighted average of γ k for k Υ except that the term 1/η in the denominator of (5) makes λ a bit larger than the actual eighted average. Hoever, for an increasing η, λ converges to the exact value of the eighted average specified above. Hereafter, e propose a recursive Grab-n-Pull procedure to fully determine Υ, hile e can obtain λ via (4). The proposed approach ill make use of the folloing observation: Lemma. If γ i < λ for any i [K], then i Υ. Proof. The inequality γi < λ implies that λ i γi. Considering the discussion above (3), one can conclude that λ i λ, hich due to the definition of λ yields λ i = λ. Hence, the proof is complete. Without loss of generality, and for the sake of simplicity, e assume in the sequel that the matrix pairs (A i, B i)} are sorted in such a ay to form the ascending order: γ 1 γ γ K. (6) Based on the above ordering, the Grab-n-Pull approach is described in Table 1. Moreover, an illustration of the method is depicted in Fig. 1. The name of the method, i.e. Grab-n-Pull, comes from the intuition that the method grabs and pulls the loest values of λ i} to a level hich is suitable for optimization of the alternative objectives, hile achieving equality, at least for the loest λ is. Due to the key role of Grab-n-Pull procedure in the proposed optimization frameork, e also use the term Grab-n-Pull hen referring to the general frameork. In the folloing, e present a specific criterion on η that not only bounds the objective of P 4 (and P 3), but also guarantees the convergence of the proposed approach. Theorem 1. Suppose η is chosen such that η > η lb 1 P ( max σ 1 min(b k ) }). (7) Then, f(λ) is upper bounded (at its maximizer λ, see (4)) as f(λ ) α k β k 1/η (8) σ max (( B k ) ) 1 1 ( )} ηp I A k. The proof of Theorem 1 is omitted herein due to lack of space. Nevertheless, to see hy the latter result implies the convergence of our algorithm, one can observe that different steps of the proposed frameork lead to an increasing objective of P 4 (and P 3). To guarantee convergence in terms of the objective value, e only need to sho that the objective is bounded from above a condition hich ill be met by satisfying (7).

4 Table 1. Recursive Grab-n-Pull Procedure to Determine Υ Step : Set Υ =. Step 1: Include 1 in Υ. Remark: Based on Lemma, the primitive index 1 belongs to Υ, as λ is alays larger than γ 1. Step : Given the current index set of minimal variables Υ, obtain λ using (4). Remark: Note that if λ is smaller than γ k for all k [K]\Υ then the obtained Υ is optimal, as all λ k ith k [K]\Υ have chosen their values freely to maximize the objective of Λ; as a result, adding other indices to Υ ill lead to a decreased objective of Λ. Step 3: Let h} [K] denote the indices for hich h / Υ. If γ h λ, include h in Υ, and goto Step ; otherise stop. Remark: This is a direct consequence of Lemma, particularly considering that λ is only increasing ith groing Υ, hich corresponds to adding larger γ i s to the eighted sum in (5). 3. A NUMERICAL EXAMPLE WITH DISCUSSIONS In this section, a brief numerical example is provided to investigate the performance of the proposed method (performing the optimization. r. t. to all variables at each iteration). To this end, e consider a donlink transmitter ith M = 4 antennas, as ell as K = 4 single-antenna users. The entries of the channel vectors h i are dran from an i.i.d. complex Gaussian distribution ith zeromean and unit-variance. The Gaussian noise components received at each user antenna are assumed to have unit variance, i.e. σ i = 1 for all i [K]. We consider a total-poer constraint ith P = 1, and stop the optimization iterations henever the objective increase becomes bounded by ɛ = 1 6. Note that hile the shado values γ i } represent the value of the fractional quadratic terms in the original objective P 1, the auxiliary variables λ i} tend to be as close as possible to γ i } depending on the eight (η) of the penalty-terms in P 3 and P 4. In Fig., e present the transition of variables γ i } and λ i} vs. the iteration number for to different settings of η; namely η = 5 and η = 1. Although ith a larger η one may expect a loer value of the penalty functions in P 3 and P 4, a loer η can play a useful role in speeding up the algorithm. Specifically, one can easily see that if η is large, the value λ in (5) ill be slightly greater than the eighted average of γ k}, hereas for smaller values of η, the bounces from the eighted average are much larger and the convergence can occur much quicker. This phenomenon can also be observed in Fig.., noting that the aforementioned values of η are chosen to accentuate the trade-off originated from the selection of η. It should be mentioned that for the special case of the totalpoer constraint, all optimal λ i} are shon to be identical [1], i.e. λ i = λ for all i [K]. This explains the identical values obtained by the method in Fig.. The results leading to Fig. ere obtained in a fe seconds on a standard PC. 4. CONCLUSION An optimization frameork for efficient precoding/beamforming in fairness-achieving netorks as proposed. Thanks to a quadratic reformulation of the original problem, the proposed method can handle different signal constraints by employing the poer method-like iterations. Various aspects of the proposed approach ere studied. Fig. 1. An illustration of the Grab-n-Pull procedure. The optimal values of λ i} are obtained hen λ < γ 3, hich sets Υ to 1, }. γ i } γ i } λi} (a) λi} (b) Fig.. Transition of the optimization parameters (distinguished by colors and line-styles) vs. the for different eights (η) of the penalty-term in P 3 and P 4: (a) η = 5, and (b) η = 1.

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