Sample Approximation-Based Deflation. Approaches for Chance SINR Constrained Joint Power and Admission Control

Transcription

1 Sample Approximation-Based Deflation 1 Approaches for Chance SINR Constrained Joint Power and Admission Control arxiv: v2 [cs.it] 21 Aug 2013 Ya-Feng Liu, Enbin Song, and Mingyi Hong Abstract In an interference limited network, joint power and admission control (JPAC) aims at supporting a maximum number of links at their specified signal to interference plus noise ratio (SINR) targets while using a minimum total transmission power. Most existing works on JPAC assume the perfect instantaneous channel state information (CSI). However, in practical wireless systems, the CSI is prone to errors, and such imperfect CSI can deteriorate the system performance drastically. Even when the perfect CSI is available, dynamic JPAC according to fast channel variations would result in high computational cost and excessive signalling overhead. In this paper, we consider the JPAC problem with imperfect CSI, that is, we assume that only the channel distribution information (CDI) is available. We formulate the JPAC into a chance (probabilistic) constrained program, where each link s SINR outage probability is enforced to be less than or equal to a specified tolerance. The chance SINR constrained JPAC formulation can maximize the number of long-term supported links by using a minimum total transmission power, and at the same time satisfy short-term SINR requirements with high probability. Unfortunately, the chance SINR constraints are computationally intractable because in general they do not have closed-form expressions and are not convex. To circumvent such difficulty, we propose to use the sample (scenario) approximation scheme to convert the chance constraints into finitely many (depending Part of this work has been presented in the 14th IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Darmstadt, Germany, June 16 19, 2013 [13]. Y.-F. Liu (corresponding author) is with the State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, , China ( yafliu@lsec.cc.ac.cn). E. Song is with the Department of Mathematics, Sichuan University, Chengdu, , China ( e.b.song@163.com). M. Hong is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA (mhong@umn.edu).

2 2 on the sample size) simple linear constraints. Furthermore, we reformulate the sample approximation of the chance SINR constrained JPAC problem as a group sparse minimization problem and then relax it to a second-order cone program (SOCP). The solution of the SOCP relaxation problem can be used to check the simultaneous supportability of all links in the network and to guide an iterative link removal procedure (deflation). Instead of solving the SOCP with general-purpose solvers like CVX, we exploit its special structure and custom-design an efficient algorithm for solving it. Finally, we illustrate the effectiveness and efficiency of the proposed sample approximation-based deflation approaches by simulations. Index Terms Chance SINR constraint, group sparse, power and admission control, sample approximation. I. INTRODUCTION Joint power and admission control (JPAC) has been recognized as an effective tool for interference management in cellular, ad hoc, and cognitive underlay wireless networks for two decades [1] [26]. The goal of JPAC is to support a maximum number of links at their specified signal to interference plus noise ratio (SINR) targets while using a minimum total transmission power when all links in the interference limited network can not be simultaneously supported. JPAC can not only determine which interfering links must be turned off and rescheduled along orthogonal resource dimensions (such as time, space, or frequency slots), but also alleviate the difficulties of the convergence of stand-alone power control algorithms. For example, a longstanding issue associated with the Foschini-Miljanic algorithm [3] is that, it does not converge when the preselected SINR levels are infeasible. In this case, a JPAC approach must be adopted to determine which links to be removed. The JPAC problem can be solved to global optimality by checking the simultaneous supportability of every subset of links. However, the computational complexity of this enumeration approach grows exponentially with the total number of links. Theoretically, the problem is known to be NP-hard to solve (to global optimality) and to approximate (to constant ratio global optimality) [1], [4], [11], so various heuristics algorithms [1] [26] have been proposed. In particular, the reference [1] proposed a convex approximation-based algorithm, called linear programming deflation (LPD) algorithm. Instead of solving the original NP-hard problem directly, the LPD algorithm solves an appropriate LP approximation of the original problem at each iteration and use its solution to guide the removal of interfering links. The removal procedure is terminated until all the remaining links in the network are simultaneously supportable. The reference [11] developed another LP approximation-based new linear programming deflation (NLPD)

3 3 algorithm for the JPAC problem. In [11], the JPAC problem is first equivalently reformulated as a sparse l 0 -minimization problem and then its l 1 -convex relaxation is used to derive a LP, which is different from the one in [1]. Again, the solution to the derived LP can guide an iterative link removal procedure, and the removal procedure is terminated if all the remaining links in the network are simultaneously supportable. Similar ideas were also used in [14], [20] to solve the joint beamforming and admission control problem for the cellular downlink network. Most of the aforementioned works on the joint power/beamforming and admission control problem such as [1], [2], [4], [10] [12], [14] (name just a few) assume the perfect instantaneous channel state information (CSI). In [1], the authors also considered the worst-case robust JPAC problem with bounded channel estimation errors. The key in [1] is that the relaxed LP with bounded uncertainty can be equivalently rewritten as a second-order cone program (SOCP). The overall approximation algorithm remains similar to LPD for the case of the perfect CSI, except that the SOCP formulation is used to carry out power control and its solution is used to check whether links are simultaneously supportable in the worst case. However, the assumption of the perfect CSI generally does not hold true due to CSI estimation errors or limited CSI feedback in practice [27], [28]. Even though the instantaneous CSI can be perfectly available, dynamic JPAC in accordance with its variations would lead to excessively high computational and signaling costs. In this paper, we consider the chance (probabilistic or outage-based) SINR constrained JPAC problem, where a random channel model is used and each link s SINR outage probability must be kept below a given tolerance. Different from most of the aforementioned works on JPAC where the perfect CSI is assumed, our new formulation only requires the availability of channel distribution information (CDI). Due to the fact that the CDI can remain unchanged over a relatively long period of time, JPAC based on the CDI can therefore be performed on a relatively slow timescale (compared to fast fluctuations of instantaneous channel conditions), hence the overall computational cost and signaling overhead can be significantly reduced. Moreover, the chance SINR constrained JPAC formulation can maximize the number of long-term supported links by using a minimum total transmission power, and at the same time guarantee that short-term SINR requirements are respected with high probability, which depends on the user-specified outage tolerance. In fact, due to the usefulness of characterizing the Quality-of-Service (QoS) constraints in terms of an outage probability, the chance constrained programming methodology has been widely applied to wireless system designs in recent years including power control [29] [33] and beamforming design [28], [34] [42]. However, as far as we know, such methodology has not been used in the context of JPAC. This is

4 4 largely due to the computational challenge of solving the chance SINR constrained JPAC problem. First, chance SINR constraints do not have closed-form expressions and are not convex in general. Second, even when the CSI is perfectly available, the JPAC problem is NP-hard to solve and to approximate [1], [4], [11]. To the best of our knowledge, this is the first paper that addresses the chance SINR constrained JPAC problem. Thanks to recent advances in chance constrained optimization [43], [44], the chance SINR constraint can be approximated via sampling with provable performance guarantee. We further show that the sample approximation of the chance SINR constrained JPAC problem can be equivalently reformulated as a group sparse minimization problem, which is a nontrivial extension of the sparse formulation in [11]. Then, we consider its convex relaxation, which is an SOCP (different from the one in [1]), and use the solution of the SOCP relaxation problem to check the simultaneous supportability of all links in the network and guide an iterative link removal procedure (deflation). Instead of relying on standard SOCP solvers to solve the derived SOCP, we exploit its special structure and custom-design an efficient algorithm for solving it. Moreover, we develop an easily checkable necessary condition for all links in the network to be simultaneously supported at their desired SINR requirements. This condition allows us to iteratively remove strong interfering links and therefore significantly accelerate the deflation process. Numerical results demonstrate that the proposed sample approximation-based deflation approaches perform very well. Notations: Denote the index set {1,2,,K} by K. Lowercase boldface and uppercase boldface are used for vectors and matrices, respectively. For a given vector x, the notations max{x}, min{x}, (x) k, and x 0 stand for its maximum entry, its minimum entry, its k-th entry, and the indicator function of x (i.e., x 0 = 0 if x = 0 and x 0 = 1 otherwise), respectively. The expression x 1 x 2 represents the Hadamard product and max{x 1,x 2 } (min{x 1,x 2 }) the component-wise maximum (minimum) of two vectors x 1 and x 2. For any subset I K, A I stands for the matrix formed by the rows of A indexed by I. We use (A 1,A 2 ) to denote the matrix formed by stacking matrices A 1 and A 2 by column and use (A 1 ;A 2 ) to denote the matrix formed by stacking A 1 and A 2 by row. Similar notations apply to stacking of vectors and scalars. Finally, we use e to represent the vector of an appropriate size with all components being one, I the identity matrix of an appropriate size, and E k the matrix of an appropriate size with all entries being zero except its k-th column entries being one, respectively. II. REVIEW OF THE NLPD ALGORITHM The algorithms developed for the chance SINR constrained JPAC problem in this work are based on the

5 5 NLPD algorithm [11] for the JPAC problem that assumes the perfect CSI. To streamline the presentation, we briefly review the NLPD algorithm in this section. The basic idea of the NLPD algorithm is to update the power and check whether all links can be supported or not. If the answer is yes, then terminate the algorithm; else drop one link from the network and update the power again. The above process is repeated until all the remaining links can be simultaneously supported. Specifically, consider a K-link (a link corresponds to a transmitter-receiver pair) interference channel with channel gains g k,j 0 (from transmitter j to receiver k), noise power η k > 0, SINR target γ k > 0, and power budget p k > 0 for k,j K. Denote the power allocation vector by p = (p 1,p 2,,p K ) T and the power budget vector by p = ( p 1, p 2,, p K ) T. Treating interference as noise, we can write the SINR at the k-th receiver as SINR k (p) = g k,k p k η k + j k g k,j p j, k K. (1) Correspondingly, we introduce an equivalent normalized channel. In particular, we use q = (q 1,q 2,,q K ) T (2) with q k = p k / p k to denote the normalized power allocation vector, and use c = (c 1,c 2,,c K ) T with c k = (γ k η k )/(g k,k p k ) > 0 to denote the normalized noise vector. We denote the normalized channel matrix by A R K K with its (k,j)-th entry 1, if k = j; a k,j = γ kg k,j p j, if k j. g k,k p k With these notations, it is simple to check that SINR k (p) γ k (Aq c) k 0. Based on the Balancing Lemma [6], we reformulate the joint power and admission control problem as a sparse optimization problem min q (c Aq) k 0 +α p T q k K s.t. 0 q e, where 0 < α < α 1 := 1/e T p. In the above, e is the all-one vector of length K. Since problem (3) is NP-hard [1], we further consider its l 1 -convex relaxation (which is equivalent to an LP; see [11]) (3) min q c Aq 1 +α p T q s.t. 0 q e. (4)

6 6 By solving (4), we know whether all links in the network can be simultaneously supported or not. If not, we drop one link (mathematically, delete the corresponding row and column of A and the corresponding entry of p and c) from the network according to some removal strategy, and solve a reduced problem (4) until all the remaining links are supported. III. PROBLEM FORMULATION Consider the chance SINR constrained JPAC problem, where the channel gains {g k,j } in the SINR expression (1) are random variables. In this paper, we assume the distribution of {g k,j } are known. However, we do not assume any specific channel distribution, which is different from most of the existing works on outage probability constrained resource allocation for wireless systems [28], [31] [36], [38] [42]. We also assume that all coordinations and computations are carried out by a central controller who knows the CDI of all links. Since {g k,j } in (1) are random variables, we need to redefine the concept of a supported link. We call link k is supported if its outage probability is below a specified tolerance ǫ (0,1), i.e., P(SINR k (p) γ k ) 1 ǫ, (5) where the probability is taken with respect to the random variables {g k,j }. The chance SINR constrained JPAC problem can be formulated as a two-stage optimization problem. Specifically, the first stage maximizes the number of admitted links: max p,s S s.t. P(SINR k (p) γ k ) 1 ǫ, k S K, (6) 0 p p. We use S 0 to denote the optimal solution for problem (6) and call it maximum admissible set. Notice that the solution for (6) might not be unique. The second stage minimizes the total transmission power required to support the admitted links: min k S {p 0 p k k} k S0 s.t. P(SINR k (p) γ k ) 1 ǫ, k S 0, (7) 0 p k p k, k S 0. Due to the choice of S 0, power control problem (7) is feasible.

7 7 IV. SAMPLE APPROXIMATION AND REFORMULATION In general, the chance SINR constrained optimization problem (6) and problem (7) are difficult to solve in an exact fashion, since it is difficult to obtain the closed-form expression of (5). In this section, we first approximate the computationally intractable chance SINR constraint via sampling, and then reformulate the sample approximation of problems (6) and (7) as a group sparse optimization problem. Three distinctive advantages of the sample approximation scheme are as follows. First, it works for general channel distribution models. Second, the sample approximation technique significantly simplifies problems (6) and (7) by replacing the difficult chance SINR constraint with finitely many simple linear constraints. Last but not the least, solving the sample approximation problem returns a solution to the original chance constrained problem with guaranteed performance [43], [44]. A. Sample Approximation { } N We handle the chance SINR constraint via sample approximations [43], [45]. Suppose gk,j n are n=1 N independent samples drawn according to the distribution of {g k,j } by the central controller, we use SINR n g k (p) := n k,k p k η k + j kg n k,j p j γ k, n N := {1,2,,N} (8) to approximate the chance SINR constraint (5). Intuitively, if the sample size N is sufficiently large, then the power allocation vector p satisfying (8) will satisfy the chance SINR constraint (5) with high probability. As shown in [30], [44], we have the following Theorem. Theorem 4.1: If N N := 1 (K 1+ln 1δ ) ǫ + 2(K 1)ln 1 1 δ +ln2 δ (9) for any δ (0,1), then the solution to problem min p e T p s.t. SINR n k (p) γ k, k K, n N, 0 p p, (10) will satisfy each of the chance SINR constraint (5) with probability at least 1 δ. Theorem 4.1 shows that the chance SINR constraint (5) will hold true with high probability if all sampled SINR constraints in (8) are satisfied. In the sequel, link k is said to be supported if all constraints in (8) are satisfied simultaneously. If problem (10) is feasible, then it has the following property.

8 8 Lemma 4.1 (Balancing Lemma): Suppose p is the solution to problem (10). Then for each k K, there must exist an index 1 n k N such that SINR nk k (p) = γ k. It is easy to show Lemma 4.1 by contradiction. Assume that there exists an index k K such that SINR n k (p) > γ k for all n N. Then we can appropriately decrease p k and strictly decrease the objective function of problem (10) without violating the other links sampled SINR constraints. B. Sampled Channel Normalization To facilitate the reformulation of the sample approximation of problems (6) and (7) and the development of efficient algorithms, we normalize the sampled channel parameters. To this end, we use ( ) T γ k η k γ k η k γ k η k c k =,,, R N 1 g 1 k,k p k g 2 k,k p k g N k,k p k to denote the normalized noise vector of link k. Define 1, if k = j; a n k,j = γ kg k,j p n j g k,k p n, if k j, k and a n k = ( a n k,1,an k,2,,an k,k) R 1 K, n N, k K, A k = ( a 1 k ;a2 k ; ) ;an k R N K, k K. Notice that the entries of the k-th column of A k are one, and all the other entries are nonpositive. This special structure of A k (k K) will play an important role in the reformulation of the sample approximation problem. Furthermore, we let and c = (c 1 ;c 2 ; ;c K ) R NK 1 A = (A 1 ;A 2 ; ;A K ) R NK K. With the above notations and (2), we can see that SINR n k (p) γ k, n N A k q c k. Consequently, the sample approximation of problem (6) can be equivalently rewritten as max q,s S s.t. A k q c k 0, k S K, (11) 0 q e,

9 9 and the sample approximation of problem (7) can be restated as min p k q k {q k} k S0 k S 0 s.t. A k q c k 0, k S 0, (12) 0 q k 1, k S 0, where S 0 is the solution of the sampled admission control problem (11). It is easy to see that the sampled power control problem (12) is feasible due to the choice of S 0. C. Group Sparse Minimization Reformulation In this subsection, we show (see Theorem 4.2) that the two-stage sampled joint power and admission control problem (11) and (12) can be reformulated as a single-stage group sparse optimization problem min max{c k A k q,0} 0 +α p T q q k K (13) s.t. 0 q e, where 0 < α < α 1 := 1/e T p. (14) Actually, if there are more than one maximum admissible set (i.e., the solution for problem (11) may not be unique), the formulation (13) is capable of picking the one with minimum total transmission power as a result of including the second term in the objective of (13). These results are formally stated in Theorem 4.2 below, and its proof is relegated to Appendix A. Theorem 4.2: Suppose q is the solution to problem (13). Then the optimal value of problem (13) lies in (M,M + 1) if and only if the optimal value of problem (11) is equal to M (i.e., there exists M K with M = M such that A k q c k for all k M). In fact, the set of links indexed by { k K A k q c k } (whose cardinality is M) is simultaneously supportable by the power allocation p = p q. Moreover, p T q is the minimum total transmission power required to support any M links in the network. The formulation (13) also has the following property stated in Proposition 4.1, which is mainly due to the special structure of A k. The proof of Proposition 4.1 can be found in Appendix B. Proposition 4.1: Suppose that q is the solution to problem (13) and link k is supported at the point q (i.e., A k q c k ). Then there must exist an index 1 n k N such that (c k A k q ) nk = 0. Proposition 4.1 implies that problem (13) can be viewed as an (nontrivial) extension of problem (3). In fact, we know from Proposition 4.1 that when N = 1, the solution of problem (13) satisfies c Aq 0

10 10 (i.e., (c Aq ) k = 0 for supported links and (c Aq ) k > 0 for unsupported links), and problem (13) reduces to problem (3). Since problem (3) is strongly NP-hard to solve to global optimality and strongly NP-hard to approximate to constant factor global optimality [1], [4], [11], it follows that problem (13) is also strongly NP-hard to solve and approximate. A main difference between problem (13) and problem (3) is as follows. In problem (3), if link k is supported, then c k A k q in this case is a scalar and equals zero; while in problem (13), if link k is supported, c k A k q is not necessarily equal to zero. This is the reason why we do not just put c k A k q in the objective of (13) but put max{c k A k q,0} there. To illustrate this, we give the following example, where K = N = 2, and A = (A 1 ;A 2 ) =, c = It is simple to check that the only possible way to simultaneously support the two links {1,2} is setting q = e. However, A 1 e c 1 = 0.3 and A 2 e c 2 = V. EFFICIENT DEFLATION APPROACHES FOR THE SAMPLED JPAC PROBLEM In this section, we develop efficient convex approximation-based deflation algorithms to solve the sampled JPAC problem (13). As we see, problem (13) has a discontinuous objective function due to the first term. However, it allows for an efficient convex relaxation. We first relax problem (13) to a convex problem, which is actually equivalent to an SOCP, and then design efficient algorithms for solving the relaxation problem. The solution to the relaxation problem can be used to check the simultaneous supportability of all links in the network and guide an iterative link removal procedure (deflation). We conclude this section with two convex approximation-based deflation algorithms for solving the sampled joint control problem (13). A. Convex Relaxation Since problem (13) is strongly NP-hard, we consider its convex relaxation problem min max{c k A k q,0} q 2 +α p T q k K s.t. 0 q e. (15)

11 11 To show the convexity of problem (15), it suffices to show that max{c k A k q,0} 2 is convex with respect to q. In fact, for any q 1,q 2, and λ (0,1), we have max{c k A k (λq 1 +(1 λ)q 2 ),0} 2 = max{λ(c k A k q 1 )+(1 λ)(c k A k q 2 ),0} 2 λmax{c k A k q 1,0}+(1 λ)max{c k A k q 2,0} 2 λ max{c k A k q 1,0} 2 +(1 λ) max{c k A k q 2,0} 2, (16) where the last inequality comes from the convexity of 2. Compared to problem (13), the objective function of problem (15) is continuous in q, but still nonsmooth. We give the subdifferential [46] of the function max{c k A k q,0} 2 in Proposition 5.1, which plays an important role in the following analysis and algorithm design. The proof of Proposition 5.1 is provided in Appendix C. Proposition 5.1: Define h k (q) = max{c k A k q,0} 2. Suppose c k A k q = 0, then h k ( q) = { A T k s s 0, s 2 1 }, Further, denoting N k + = {n (c k A k q) n > 0}, then h k ( q) = n N (c + k k A k q) n (a n k )T max{c k A k q,0} 2 = AT k max{c k A k q,0} max{c k A k q,0} 2. (17) One may ask why we do not use a smooth convex relaxation instead of (15)? The reason is that, with the use of the nonsmooth mixed l 2 /l 1 norm, the formulation (15) is able to characterize and induce the group sparsity of the vector x := max{c Aq,0} (recall that the objective of problem (13) actually is to find a feasible q such that the vector x is as sparse as possible in the group sense). To understand this, observe that the first term of the objective function of (15), the l 1 norm of the vector ( x 1 2, x 2 2,, x K 2 ) T, is a good approximation of its l 0 norm, which is equal to the l 0 norm of the vector ( x 1 0, x 2 0,, x K 0 ) T. More discussions on using the l 2 /l 1 norm to recover the group sparsity can be found in [47]. We now discuss the choice of the parameter α in (15). The parameter α in (15) should be chosen appropriately such that the following Never-Over-Removal property is satisfied: the solution of problem (15) should simultaneously support all links at their desired SINR targets with minimum total transmission power as long as all links in the network are simultaneously supportable. Otherwise, since the solution of (15) will be used to check the simultaneous supportability of all links and guide the links removal, it may mislead us to remove the links unnecessarily. Theorem 5.1 gives an upper bound of the parameter α

12 12 to guarantee the Never-Over-Removal property. The proof of Theorem 5.1 (see Appendix D) is mainly based on Lemma 4.1 and Proposition 5.1. Theorem 5.1: Suppose there exists some vector q such that 0 q e and Aq c. Then the solution of problem (15) with 0 α α 2 := min{c} Kmax{ p} can simultaneously support all links at their desired SINR targets with minimum total transmission power. Combining (14) and (18), we propose to choose the parameter α in (15) according to where c 1,c 2 (0, 1) are two constants. (18) α = min{c 1 α 1, c 2 α 2 }, (19) Link Removal Strategy. The solution of problem (15) can be used to guide the link removal process. In particular, by solving (15) with α given in (19), we know whether all links in the network can be simultaneously supported by simply checking if its solution q satisfies A q c. Furthermore, if all links in the network can not be simultaneously supported, we need to remove at least one link from the network. In particular, picking the worst sampled channel index n k = argmax{c k A k q} and letting q e k = max{c k A k q} for k K, we may remove the link with the largest interference plus noise footprint [1] k = argmax a nk k,j q j + a nj j k j k j,k q k +c nk k the link with the largest excess transmission power [4] k = argmax a nk k,j qe j + a nj j k j k or the link with the maximum feasibility violation j,k qe k, (20), (21) k = argmax{ max{c k A k q} 2 }. (22) We remark that the above link removal strategies (20) and (21) are different from the ones given in [1], [4] due to the presence of channel normalization. We compared the three removal strategies (20), (21), and (22) in our simulations, and found that the first one works slightly better than the second one (in terms of the number of supported links and the total transmission power), and both of them works much better than the last one. Therefore, we shall use the removal strategy (20) in the proposed deflation algorithms. In the next subsection, we design efficient algorithms to solve the convex but nonsmooth problem (15).

13 13 B. Solution for Relaxation Problem (15) By introducing auxiliary variables x = (x 1 ;x 2 ; ;x K ) and t = (t 1 ;t 2 ; ;t K ), problem (15) can be transformed into the following SOCP min q,x,t t k +α p T q k K s.t. x k t k, k K, c Aq x, 0 x, 0 q e, which is convex and smooth and can be solved by using the standard solver like CVX [48]. Problem (15) can also be solved directly by the subgradient projection method [49]. The subgradient of the objective of problem (15) can be easily obtained from Proposition 5.1, and the projection of a point q onto the box Q = {q 0 q e} admits a closed-form solution, i.e., Proj Q (q) = min{e,max{q,0}}. With an appropriate step size rule, the subgradient projection method is globally convergent. However, it has a slow convergence rate that is typically sublinear O ( 1/ t ), where t is the iteration index, and is sensitive to the choice of the initial step size. Next, we develop a custom-design algorithm for problem (15) by first smoothing the problem and then applying the efficient projected alternate Barzilai-Borwein (PABB) algorithm [50], [51] to solve its smooth counterpart. More specifically, we smooth problem (15) by min q f µ (q) = k K s.t. 0 q e, max{c k A k q,0} 2 2 +µ2 +α p T q where µ > 0 is the smoothing parameter. By (17) in Proposition 5.1, the objective function f µ (q) of problem (24) is differentiable everywhere and its gradient is given by f µ (q) = k K A T k max{c k A k q,0} max{c k A k q,0} 22 +µ2 +α p. (23) (24) As the parameter µ tends to zero, f µ (q) uniformly approaches the objective function in (15) and hence the solution of the smoothing problem (24) converges to the solution of problem (15).

14 14 Now we apply the PABB algorithm [50], [51] to solve the smoothing problem (24). The projected Barzilai-Borwein algorithm is developed to solve the optimization problem min f(x) x s.t. x X, where f(x) is differentiable and the projection onto the feasible set X is relatively easy to compute. At the t-th iteration, the algorithm updates the design variable x by where the step size α BB (t) is either x(t+1) = Proj X [x(t) α BB (t) x f(x(t))], α BB1 (t) = s(t)t s(t) s(t) T y(t) or α BB2 (t) = s(t)t y(t) y(t) T y(t) with s(t) = x(t) x(t 1) and y(t) = x f(x(t)) x f(x(t 1)). In particular, the PABB algorithm alternatingly uses the large step size α BB1 and the small step size α BB2. When using the PABB algorithm to solve problem (24), we employ the continuation technique [52], [53]. That is, instead of using a tiny µ, we solve (24) with a series of values for µ, which are gradually decreasing, to obtain an approximate solution of (15). It turns out the continuation technique can reasonably improve the computational efficiency. (25) C. Convex Approximation-Based Deflation Algorithms The basic idea of the proposed convex approximation-based deflation algorithm for the sampled JPAC problem (13) is to solve the power control problem (15) and check whether all links can be supported or not; if not, remove a link from the network, and solve a reduced problem (15) again until all the remaining links are supported. As in [11], to accelerate the deflation procedure (avoid solving too many optimization problems in the form of (15)), we derive an easy-to-check necessary condition for all links in the network to be simultaneously supported. If all links can be simultaneously served, there exists a vector q such that 0 q e and Aq c. By the definition of A, we have q (max{c 1 };max{c 2 }; ;max{c K }) := c max. Define µ = A T e

15 15 and set µ + = max{µ,0} and µ = max{ µ,0}. It is obvious that µ = µ + µ. Multiplying e T from both sides of Aq c, we get that (µ + µ ) T q e T c. Moreover, we can obtain µ T + e µt + q µt q+et c µ T cmax +e T c, where the first inequality is due to q e and the last one is due to q c max. Therefore, the condition is necessary for all links to be simultaneously supported. µ T +e ( µ T c max +e T c ) 0 (26) We can use the necessary condition (26) in the link removal process. In particular, if (26) is violated, we should drop at least one link from the network. We propose to remove the link k 0 according to k 0 = argmax ā k K k,j + ā j,k + c k, (27) j k j k which corresponds to applying the SMART rule [4] to the normalized sampled channel and substituting q = e. In (27), ā k,j and c k are the averaged sample channel gain and noise, i.e., ā k = (ā k,1,ā k,2,,ā k,k ) = et A k N, c k = et c k N, k K. The proposed convex approximation-based deflation algorithm framework for problem (13) is described in Algorithm 1. In the framework, if the power control problem (15) is solved via solving its equivalent SOCP reformulation (23), we call the corresponding algorithm SOCP-D; while if problem (15) is solved via using the PABB algorithm to solve its smoothing counterpart (24), we call the corresponding algorithm PABB-D. The SOCP-D algorithm is of polynomial time complexity, i.e., it has a complexity of O(N 0.5 K 3.5 (N +K)), since it needs to solve K SOCP problems (23) and solving one SOCP problem in the form of (23) requires O(N 0.5 K 2.5 (N +K)) operations [54, Page 423]. We should remark that the postprocessing step (Step 5) in the framework aims at admitting the links removed in the preprocessing and admission control steps. Specifically, we enumerate all the removed links and admit one of them if it can be supported simultaneously with the already supported links. If there are more than one such candidates, we pick the one such that the minimum total transmission power is needed to simultaneously support it with the already supported links. The postprocessing step is terminated if no such candidate exists. The specification of the postprocessing step can be found in Appendix E.

16 16 Algorithm 1: A Convex Approximation-Based Deflation Algorithm Framework Step 1. Initialization: Input data (A,c, p). Step 2. Preprocessing: Remove link k 0 iteratively according to (27) until condition (26) holds true. Step 3. Power control: Compute parameter α by (19) and solve problem (15); check whether all links are supported: if yes, go to Step 5; else go to Step 4. Step 4. Admission control: Remove link k 0 according to (20), set K = K\{k 0 }, and go to Step 3. Step 5. Postprocessing: Check the removed links for possible admission. VI. NUMERICAL SIMULATIONS To illustrate the effectiveness and efficiency of the two proposed convex approximation-based deflation algorithms (SOCP-D and PABB-D), we present some numerical simulation results in this section. The number of supported links, the total transmission power, and the execution CPU time are used as the metrics for comparing different algorithms. Simulation Setup: As in [1], each transmitter s location obeys the uniform distribution over a 2 Km 2 Km square and the location of each receiver is uniformly generated in a disc with center at its corresponding transmitter and radius 400 m, excluding a radius of 10 m. Suppose that the channel coefficient h k,j is generated from the Rician channel model [55], i.e., ( ) κ 1 h k,j = κ+1 + κ+1 ζ 1 k,j, k, j K, (28) where ζ k,j obeys the standard complex Gaussian distribution, i.e., ζ k,j N(0,1), d k,j is the Euclidean distance from the link of transmitter j to the link of receiver k, and κ is the ratio of the power in the line of sight (LOS) component to the power in the other (non-los) multipath components. For κ = 0 we have Rayleigh fading and for κ = we have no fading (i.e., a channel with no multipath and only a LOS component). The parameter κ therefore is a measure of the severity of the channel fading: a small κ implies severe fading, a large κ implies relatively mild fading. The channel gain {g k,j } are set to be: d 2 k,j g k,j = h k,j 2, k, j K. In addition, each link s SINR target is set to be γ k = 2 db ( k K), each link s noise power is set to be η k = 90 db ( k K), and the power budget of the link of transmitter k is set to be p k = bp k, k K, (29)

17 17 where p k is the minimum power needed by link k to meet its SINR requirement in the absence of any interference from other links when κ = + in (28). Benchmark: If κ = +, then there is no uncertainty of channel gains, and the number of supported links in this case should be greater than or equal to the number of supported links under the same channel conditions except where κ < +. In addition, if the number of supported links under these two cases are equal to each other, the total transmission power in the former channel condition should be less than the one in the latter channel condition. In fact, if κ = +, the corresponding JPAC problem (13) reduces to problem (3), which can be solved efficiently by the NLPD algorithm in [11]. The solution given by the NLPD algorithm will be used as the benchmark to compare with the two proposed algorithms, since the NLPD algorithm was reported to have the close-to-global-optimal performance in [11]. Choice of Parameters: We set the parameters ǫ, δ, and K in (9) to be 0.1, 0.05, and 10, respectively. We remark that K in equation (9) is the number of supported links but not the number of total links. Substituting these parameters in (9), we obtain N = 200, and we set N = 200 in all of our simulations. Both of the parameters c 1 and c 2 in (19) are set to be When applying the PABB algorithm with the continuation technique to solve the power control problem (15), we always set the smoothing parameter µ to be 1 at first, and then gradually decrease it by multiplying a factor of 0.1 until it becomes less than We do three sets of numerical experiments, where one is (κ,b) = (+,2), one is (κ,b) = (100,4), and another one is (κ,b) = (10,30). Finally, we use CVX [48] to solve the SOCP problems in the SOCP-D algorithm. Simulation Results and Analysis: Table I summarizes the statistics of the number of supported links of 200 Monte-Carlo runs in three sets of numerical experiments. For instance, 678 = in the fourth column of Table I stands for that when K = 4, total 678 links are supported in these 200 Monte-Carlo runs, and amongest them, 2 links are supported 13 times, 3 links are supported 96 times, and 4 links are supported 91 times. Figs. 1, 2, and 3 are obtained by averaging over the 200 Monte-Carlo runs. They plot the average number of supported links, the average total transmission power, and the average execution CPU time of the proposed SOCP-D and PABB-D algorithms and the benchmark versus different number of total links, respectively. It can be seen from Fig. 1 that the number of supported links by the two proposed algorithms (for fading channels) is less than the benchmark (for deterministic channels). This shows that the uncertainty of channel gains could lead to a (significant) reduction in the number of supported links. This can be clearly observed from Table I. For instance, when K = 4, all links can be simultaneously supported 91 times when κ = +, 84 times when κ = 100, and only 57 times when κ = 10. In fact, this is the

18 18 TABLE I STATISTICS OF THE NUMBER OF SUPPORTED LINKS OF 200 MONTE-CARLO RUNS. Number of Total Links Parameters (κ, b) Algorithm Statistics of the Number of Supported Links 4 (+, 2) Benchmark 678=2*13+3*96+4*91 4 (100, 4) SOCP-D/PABB-D 671=2*13+3*103+4*84 4 (10, 30) SOCP-D/PABB-D 629=1*1+2*26+3*116+4*57 12 (+, 2) Benchmark 1455=5*6+6*34+7*81+8*60+9*16+10*3 12 (100, 4) SOCP-D/PABB-D 1408=5*10+6*50+7*75+8*54+9*9+10*2 12 (10, 30) SOCP-D/PABB-D 1224=4*8+5*48+6*69+7*64+8*9+9*2 20 (+, 2) Benchmark 1951=7*7+8*26+9*51+10*63+11*34+12*17+13*1+14*1 20 (100, 4) SOCP-D/PABB-D 1885=7*10+8*39+9*61+10*49+11*31+12*8+13*1+14*1 20 (10, 30) SOCP-D/PABB-D 1609=5*5+6*20+7*45+8*58+9*43+10*24+11*2+12*3 28 (+, 2) Benchmark 2339=8*3+9*17+10*29+11*43+12*46+13*35+14*16+15*9+16*1+17*1 28 (100, 4) SOCP-D/PABB-D 2255=8*9+9*18+10*38+11*46+12*46+13*26+14*10+15*7 28 (10, 30) SOCP-D/PABB-D 1953=6*2+7*10+8*32+9*46+10*46+11*40+12*14+13*7+14*3 12 Number of Supported Links Benchmark SOCP D PABB D κ=100,b=4 κ=10,b= Number of Total Links Fig. 1. Average number of supported links versus the number of total links. reason why we associate different κ with different b in our simulations. We expect that a large b and thus large power budgets p k (cf. (29)) can compensate the performance degradation of the number of supported links caused by the large uncertainty of channel gains. Table I, Fig. 1, and Fig. 2 show that the two proposed algorithms always return the same solution to the sampled JPAC problem (13), i.e., supporting same number of links with same total transmission power. However, Fig. 3 shows that the PABB-D algorithm substantially outperforms the SOCP-D algorithm in

19 Average Total Transmission Power κ=10,b=30 κ=100,b=4 SOCP D PABB D Benchmark Number of Total Links Fig. 2. Average total transmission power versus the number of total links. 60 Average Execution Time (s) κ=10,b=30 SOCP D PABB D Benchmark κ=100,b= Number of Total Links Fig. 3. Average execution time (in seconds) versus the number of total links. terms of the average CPU time. This is not surprising, since in each power control step (i.e., solving problem (15)), the custom-design algorithm is used to carry out power control in the PABB-D algorithm while a general-purpose solver CVX is used to update power in the SOCP-D algorithm. By comparing the two sets of numerical experiments where (κ,b) = (100,4) and (κ,b) = (10,30), it can be observed from Figs. 1 and 2 that more links can be supported with significantly less total transmission power in the former case than the latter case. This is because the uncertainty of the channel gains with κ = 100 is generally smaller than the one with κ = 10. We also point out that the execution CPU time of the two proposed deflation algorithms mainly depends on how many times the power control

20 20 problem (15) is solved. In general, the larger number of links are supported, the smaller number of links are removed from the network and the smaller number of power control problems in form of (15) are solved. Therefore, the average CPU time of the proposed algorithms when (κ,c) = (10,30) is larger than the one when (κ,c) = (100,4); see Fig. 3. VII. CONCLUSIONS In this paper, we have considered the chance SINR constrained JPAC problem, and have proposed two sample approximation-based deflation approaches for solving the problem. We first approximated the computationally intractable chance SINR constraints by sampling, and then reformulated the sampled JPAC problem as a one-stage group sparse minimization problem. Furthermore, we relaxed the NP-hard group sparse minimization problem to a convex problem (equivalent to an SOCP) and used its solution to check the simultaneous supportability of all links and guide an iterative link removal procedure (deflation), resulting in efficient deflation algorithms (SOCP-D and PABB-D). The proposed approaches are particularly attractive for practical implementations for the following reasons. First, the two proposed approaches only require the CDI, which is more practical than most of the existing algorithms for JPAC where the perfect instantaneous CSI is required. Second, the two proposed approaches enjoy a low computational complexity. The SOCP-D approach has a polynomial time complexity. To further improve the computational efficiency, the special structure of the SOCP relaxation problem is exploited, and an efficient algorithm, PABB-D, is custom designed for solving it. The PABB-D algorithm significantly outperforms the SOCP-D algorithm in terms of the execution CPU time. Finally, our simulation results show that the proposed approaches are very effective by using the NLPD algorithm as the benchmark. ACKNOWLEDGMENT The first author wishes to thank Professor Yu-Hong Dai of Chinese Academy of Sciences for many useful discussions. APPENDIX A Proof of Theorem 4.2: It is simple to see the equivalence between problem (11) and the intermediate problem min q max{c k A k q,0} 0 k K (30) s.t. 0 q e

21 21 in the sense that the optimal value of problem (11) is M if and only if the minimum value of problem (30) is K M. We now establish the equivalence between problems (13) and (30) (in the sense of supporting the maximum number of links). We claim that the optimal value of (30) is M if and only if the optimal value of (13) is in the interval (M,M +1). We argue the only if and if directions separately. Recall the fact 0 < α < α 1 = 1/ p T e (cf. (14)) which implies α p T q < 1, for any 0 q e. Consequently, the total contribution from the second term in the objective function of (13) cannot exceed 1, regardless of the power allocation q. This immediately shows that if the optimal value of (30) is M, then the optimal value of (13) must be in the interval (M,M +1). To argue the converse, we note that the function k K max{c k A k q,0} 0 is discontinuous with an increment of 1, implying that max{c k A k q,0} 0 +α p T q k K k K max{c k A k q,0} 0 (31) M +1 > M +α p T q, for any choice of 0 q e, as long as k K max{c k A k q,0} 0 > M. Thus, the global minimum of (13) must be achieved at a power allocation for which max{c k A k q,0} 0 = M k K holds, i.e., k K max{c k A k q,0} 0 is fully minimized by (30). This establishes the equivalence between (30) and (13). Finally, if there are multiple sets of M links that are simultaneously supportable, then they all induce the same objective value in the first term of the objective function in (13). In this case, the second term (i.e., p T q) will play the role to select the one set of M links which requires the least amount of total transmission power. In other words, p T q = e T p is the minimum total power required to support any M links in the network. APPENDIX B Proof of Proposition 4.1: We prove Proposition 4.1 by contraction. Assume that link k is supported in (13) but A k q > c k holds true. Then we can construct a feasible point ˆq satisfying k A kˆq,0} 0 +α p k K max{c Tˆq < max{c k A k q,0} 0 +α p T q. (32) k K

22 22 Define ˆq = (ˆq 1,ˆq 2,, ˆq K ) T with max{(e k A k )q +c k }, if j = k; ˆq j = qj, if j k. Recalling the definitions of E k and A k, we know E k A k is a nonnegative matrix, and thus Since ˆq k = max{(e k A k )q +c k } > 0. A k q = E k q (E k A k )q = q k e (E k A k )q > c k, it follows that q k > max{(e k A k )q +c k } = ˆq k. Hence, ˆq is feasible (i.e., 0 ˆq q e and the inequality ˆq q holds true strictly for the k-th entry) and p Tˆq < p T q. (33) Moreover, it follows from the definition of ˆq k that A kˆq c k. For any j k, if A j q c j, we have A jˆq c j A j q c j 0, where the first inequality is due to the fact that all entries of A j except its j-th column are nonpositive and the fact ˆq q. Consequently, there holds J ˆ J, where J = {j A j q c j } and ˆ J = {j A jˆq c j }. Thus, we have k K max{c k A k q,0} 0 = k/ J max{c k A k q,0} 0 k/ ˆ J max{c k A kˆq,0} 0 (34) = k K max{c k A kˆq,0} 0. Combining (33) and (34) yields (32), which contradicts the optimality of q. This completes the proof of Proposition 4.1. APPENDIX C To prove Proposition 5.1, we first consider the simple case where h(q) = max{q,0} 2. Lemma C.1: Suppose h(q) = max{q,0} 2. Then h(0) = {s s 0, s 2 1}. If there exists i such that (q) i > 0, then h(q) is differentiable and h(q) = max{q,0} max{q,0} 2. (35)

23 23 Proof: Define S = {s s 0, s 2 1}. We claim h(0) = S. On one hand, taking any s S, we have that h(q) = max{q,0} 2 s T max{q,0} s T q = h(0)+s T (q 0), q, where the first inequality is due to the Cauchy-Schwarz inequality and the fact s 2 1, and the second inequality is due to s 0. This shows that S h(0) according to the definition of h(0) [46]. On the other hand, to show h(0) S, it suffices to show that any point s / S is not a subgradient of h(q) at point 0. In particular, if s 2 > 1, then h(q) = max{q,0} 2 q 2 = 1 < s 2 = s T q = h(0)+s T (q 0) with q = s/ s 2. Thus, the subgradient s of h(q) at point 0 must satisfy s 2 1. If s 2 1 but (s) 1 < 0 (without loss of generality), we test the point q = ( 1,0,,0) T, and obtain h(q) = max{q,0} 2 = 0 < (s) 1 = s T q = h(0)+s T (q 0). Consequently, the subgradient s of h(q) at point 0 must satisfy s 2 1 and s 0. Hence, h(0) = {s s 0, s 2 1}. Next, we show that h(q) is differentiable at the point q which has at least one positive entry, and the corresponding gradient is given in (35). In fact, although the function max{q, 0} is nondifferentiable at point q = 0, its square f(q) = (max{q,0}) 2 is differentiable everywhere; i.e., 0, if q 0; f (q) = 2q, if x > 0. According to the composite rule of differentiation, we know that the gradient of h(q) is given by (35). This completes the proof of Lemma C.1. Equipping with Lemma C.1, we now can prove Proposition 5.1. Assume that c k A k q = 0. Then for any q and any s satisfying s 2 1 and s 0, we have h k (q) = max{c k A k q,0} 2 s T (c k A k q) (from Lemma C.1) = s T (A k q A k q) = ( A T k s) T (q q) (36) = max{c k A k q,0} 2 + ( A T k s) T (q q) = h k ( q)+ ( A T k s) T (q q),

24 24 which shows that all vectors in { A T k s s 0, s 2 1 } are subgradients of h k (q) at point q. In the same way as in the proof of Lemma C.1, we can show that ifsdoes not satisfy s 2 1 ands 0, the inequality in (36) will violate for some special choices of q. Hence, h k ( q) = { A T k s s 0, s 2 1 }. If N + k, we know from the composite rule of differentiation and Lemma C.1 that h k ( q) is differentiable and its gradient is given by h k ( q) = n N (c + k k A k q) n (a n k )T max{c k A k q,0} 2 This completes the proof of Proposition 5.1. = AT k max{c k A k q,0} max{c k A k q,0} 2. APPENDIX D Proof of Theorem 5.1: Suppose all links in the network can be simultaneously supported (i.e., there exists 0 q e satisfying Aq c) and q is the solution to problem min q p T q s.t. Aq c 0, 0 q e. (37) To prove Theorem 5.1, it suffices to show that q is also the solution to problem (15) with α [0,α 2 ]. Moreover, to show q is the solution to problem (15), we only need to show that the subdifferential of the objective function of problem (15) at point q contains 0 [46]. Next, we claim the latter is true. We first characterize the subdifferential of the objective function of problem (15) at point q. It follows from Lemma 4.1 that there exists I = {n 1,n 2,,n K } such that q is the solution to the following linear system A I q := [ a n1 1 ;an2 2 ] ( ; ;ank K q = c n 1 1 ;cn2 2 ) ; ;cnk K := ci. Recalling the definition of a n k (see Subsection IV-B), we know that I A I is a nonnegative matrix. Moreover, from [56, Theorem 1.15], A I is nonsingular, A 1 I is nonnegative, and 0 < e T A 1 I c I = e T q K. (38) Define h k (q) = max{c k A k q,0} 2 for k K. It follows from [46, Theorem 23.8] that the subdifferential of the objective function of problem (15) at point q is given by { } g k +α p g k h k ( q), k K. k K

25 25 According to Proposition 5.1, h k ( q) contains 1 all vectors in { ( S k = s k a n k ) T k 0 sk 1}. (39) Therefore, all vectors in S = { A T Is+α p 0 s e } = { k Ks k ( a n k k are subgradients of the objective function of problem (15) at point q. ) T +α p 0 sk 1,k K If 0 S, the subdifferential of the objective function of problem (15) at point q contains 0 [46], which completes the proof of Theorem 5.1. In fact, consider the vector s = αa T I p. It is a nonnegative vector (since A 1 I is nonnegative), and each of its entries is less than or equal to 1 as long as α α 2. This is because e T s = e T αa T I p αmax{ p}e T A T I e α max{ p} min{c I } ct I A T I e (a) α max{ p} min{c} K 1, where (a) is due to (38) and the fact min{c} min{c I }. Substituting s = αa T I p into S, we obtain ( ) A T I s+α p = AT I αa T I p +α p = 0. Thus, 0 S. } (40) 1 The subdifferential of h k (q) at point q is not necessarily equal to S k in (39). This is because that some other entries (except the n k -th entry) of the vector c k A k q might also be zero.