Abstract:
A monotonic optimization method is for achieving the maximum weighted sum-rate in multicell downlink MISO systems. It belongs to the wireless communications field. The weighted sum-rate maximization is viewed as a monotonic optimization problem over the achievable rate region. A sensible search scheme, a sequential partition method as well as a vertex relocation method are designed to reduce the complexity and accelerate the convergence.

Description:
CROSS REFERENCE OF RELATED APPLICATION 
       [0001]    This is a U.S. National Stage under 35 U.S.C 371 of the International Application PCT/CN2012/001585, filed Nov. 26, 2012, which claims priority under 35 U.S.C. 119(a-d) to CN 201210359384.7, filed Sep. 24, 2012. 
     
    
     BACKGROUND OF THE PRESENT INVENTION 
       [0002]    1. Field of Invention 
         [0003]    The present invention describes a monotonic optimization method for achieving the maximum weighted sum-rate in multicell downlink MISO systems, which is a technique of wireless communications. 
         [0004]    2. Description of Related Arts 
         [0005]    Coordinated multicell downlink transmission is a promising technique for suppressing co-channel interference and improving the system performance. Among the different kinds of design criteria, weighted sum-rate maximization is an active research topic, which has attracted significant attention from both academics and industries. However, achieving the global optimum is very difficult as the capacity region remains unknown. A more pragmatic method that adopts single-user detection by simply treating interference as noise has stimulated some efficient algorithms. 
         [0006]    In the conventional techniques, the reference L. Venturino, N. Prasad, and X. Wang, “ Coordinated linear beamforming in downlink multi - cell wireless networks,” IEEE Transactions on Wireless Communications , vol. 9, no. 4, pp. 1451-1461, April 2010, presents an iterative coordinated beamforming algorithm to solve the Karush-Kuhn-Tucker conditions of the weighted sum-rate maximization problem. 
         [0007]    The reference Q. J. Shi, M. Razaviyayn, Z. Q. Luo, and C. He, “ An iteratively weighted MMSE approach to distributed sum - utility maximization for a MIMO interfering broadcast channel,” IEEE Transactions on Signal Processing , vol. 59, no. 9, pp. 4331-4340, September 2011, proposes a weighted minimum mean square error method which iteratively update the transmitter, the receiver and the weight matrix. 
         [0008]    However, the achievable rate region under single-user detection remains non-convex. For MISO systems, the reference R. Zhang and S. G. Cui, “ Cooperative interference management with MISO beamforming,” IEEE Transactions on Signal Processing , vol. 58, no. 10, pp. 5454-5462, October 2010, proves that beamforming techniques can achieve the Pareto-optimal points and describes a method to characterize the Pareto-boundary of the achievable rate region. Although there are already some techniques obtaining Pareto-optimal solutions, it is still a big challenge to search for the global optimality. 
         [0009]    The reference L. Liu, R. Zhang, and K. Chua, “ Achieving global optimality for weighted sum - rate maximization in the K - user Gaussian interference channel with multiple antennas,” IEEE Transactions on Wireless Communications , vol. 11, no. 5, pp. 1933-1945, 2012, views the weighted sum-rate maximization as a monotonic optimization problem and adopts the outer polyblock approximation algorithm combined with the rate profile technique to approaching the global optimal solution. To improve the efficiency of the monotonic optimization, another reference E. Bjornson, G. Zheng, M. Bengtsson, and B. Ottersten, “ Robust monotonic optimization framework for multicell MISO systems,” IEEE Transactions on Signal Processing , vol. 60, no. 5, pp. 2508-2523, 2012, provides a branch-reduce-and-bound based method. Although it converges faster than the outer polyblock approximation algorithm, it still needs too many iterations as the number of users increase. 
       SUMMARY OF THE PRESENT INVENTION 
       [0010]    Based on the conventional techniques, the present invention proposes a novel monotonic optimization method to achieve the maximum weighted sum-rate in multicell downlink MISO systems. The present invention utilizes a sensible search scheme, a sequential partition method and a vertex relocation method to accelerate the convergence. 
         [0011]    The present invention is implemented as the following steps of: 
         [0012]    Step 1: setting system parameters comprising: a cell/base station number M, a user number in an m-th cell K m , an antenna number of an m-th base station T m , a maximum transmission power of the m-th base station P m , wherein m=1, a 1 by T n  channel vector from a n-th base station to an m k -th user (i.e. the k-th user in the m-th cell) h m     k     ,n , wherein m,n=1, . . . , M, k=1, . . . , K m , a variance of a zero-mean complex Gaussian additive noise at the m k -th user σ m     k     2 , a weight of the m k -th user α m     k   , wherein m=1, . . . , M, k=1, . . . , K m ; 
         [0013]    Step 2: defining R m     k    as a rate of the m k -th user, 
         [0000]    
       
         
           
             
               R 
               
                 m 
                 k 
               
             
             = 
             
               
                 log 
                 2 
               
               ( 
               
                 1 
                 + 
                 
                   
                     
                        
                       
                         
                           h 
                           
                             
                               m 
                               k 
                             
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                     + 
                     
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                         m 
                         k 
                       
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         [0000]    wherein w m     k    is the T m  by 1 beamformer for the m k -th user, wherein m=1, . . . , M, k=1, . . . , K m ; 
         [0014]    formulating a achievable rate vector 
         [0000]    
       
         
           
             r 
             = 
             
               
                 ( 
                 
                   
                     R 
                     
                       1 
                       1 
                     
                   
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                   … 
                    
                   
                       
                   
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                       1 
                       
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                         1 
                       
                     
                   
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                       M 
                       
                         K 
                         M 
                       
                     
                   
                 
                 ) 
               
               ∈ 
               
                 ℝ 
                 + 
                 K 
               
             
           
         
       
     
         [0000]    as well as a weighted sum-rate function 
         [0000]    
       
         
           
             
               
                 f 
                  
                 
                   ( 
                   r 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     m 
                     , 
                     k 
                   
                 
                  
                 
                   
                     α 
                     
                       m 
                       k 
                     
                   
                    
                   
                     R 
                     
                       m 
                       k 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    wherein ε           +   K  indicates that r is a positive real vector of a length K with K=Σ m=1   M K m ; 
         [0015]    Step 3: defining a box [a,b]={xε           +   K |a≦x≦b}; initializing the set of boxes as ={[0,b 0 ]}; assuming K 0 =0, a (Σ i=0   m−1 K i +k)-th element (associated with the m k -th user) of b 0  is log 2 (1+P m νh m     k     ,m   H ∥ 2 /σ m     k     2 ), wherein m=1, . . . , k=1, . . . , K m ; setting a termination accuracy η and a line search accuracy δ; 
         [0016]    Step 4: initializing an upper bound f max  and a lower bound f min  of the weighted sum-rate, wherein 
         [0000]        f   max   =f ( b   0 ),  f   min =max(α· b   0 )
 
         [0000]    wherein α is a weight vector comprising weights for all the users, i.e. 
         [0000]    
       
         
           
             
               α 
               = 
               
                 
                   ( 
                   
                     
                       α 
                       
                         1 
                         1 
                       
                     
                     , 
                     … 
                      
                     
                         
                     
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                           1 
                         
                       
                     
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                           M 
                         
                       
                     
                   
                   ) 
                 
                 ∈ 
                 
                   ℝ 
                   + 
                   K 
                 
               
             
             ; 
           
         
       
     
         [0017]    Step 5: choosing a box [a,b] from that satisfies f(b)=f max , and then checking feasibility of a: whether a locates in a achievable rate region or not; wherein the feasibility is determined via a problem φ(a); 
         [0018]    the specified problem φ(a) is:
       maximize 0   subject to √{square root over (β m     k   )}∥A m     k   x+n m     k   ∥≦√{square root over (1+β m     k   )}(h m     k     ,m S m     k   x),∀m,k   p T x=0   ∥G m x∥≦√{square root over (P m )},∀m
 
with the following notations
       
 
         [0000]    
       
         
           
             
               
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              
             
                 
             
              
             
               
                 G 
                 m 
               
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                         m 
                         1 
                       
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                         m 
                         
                           K 
                           m 
                         
                       
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                 H 
               
             
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                   k 
                 
               
               = 
               
                 
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                       k 
                     
                   
                 
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               p 
               = 
               
                 
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                        
                       0 
                     
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                   ] 
                 
                 T 
               
             
           
         
       
     
         [0000]    wherein σ m     k    is a standard deviation of the zero-mean complex Gaussian additive noise at the m k -th user, I T     m    denotes a identity matrix with dimension T m ; 
         [0023]    Step 6: if the problem φ(a) is feasible, conducting a sensible search scheme for the box [a,b] to obtain a partition point r; 
         [0024]    wherein if the problem φ(a) is infeasible, updating the box set as =\[a,b] and calculating the upper bound f max =max [a,b] εf(b), then going back to the Step 5; 
         [0025]    the sensible search scheme is: 
         [0026]    denoting l ab  as the line connecting a and b; finding the intersection point c on the hyperplane {r|f(r)=f min } with the line l ab , i.e. 
         [0000]    
       
         
           
             c 
             = 
             
               a 
               + 
               
                 
                   ( 
                   
                     b 
                     - 
                     a 
                   
                   ) 
                 
                 × 
                 
                   
                     
                       f 
                       min 
                     
                     - 
                     
                       f 
                        
                       
                         ( 
                         a 
                         ) 
                       
                     
                   
                   
                     f 
                      
                     
                       ( 
                       
                         b 
                         - 
                         a 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0027]    checking feasibility of c via the problem φ(a) in the Step 5 with a=c; 
         [0028]    wherein if φ(c) is feasible, a bisection line search is conducted along the line l cb  to obtain an intersection point on a Pareto-boundary; given a line search accuracy δ, two points r min  and r max  are acquired; the partition point is set as r=r max  while the lower bound is updated as f min =f(r min ); 
         [0029]    if φ(c) is infeasible, just set the partition point as r=c; 
         [0030]    Step 7: based on the partition point r, dividing the box [a,b] into K new boxes [a (i) ,b (i) ], i=1, . . . , K using a sequential partition method; then updating the boxes set as 
         [0000]    
       
         
           
             = 
             
               
                 \[ 
                  
                 
                   a 
                   , 
                   b 
                 
                 ] 
               
               ⋃ 
               
                 { 
                 
                   
                     ⋃ 
                     
                       
                         i 
                         = 
                         1 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       K 
                     
                   
                    
                   
                     [ 
                     
                       
                         a 
                         
                           ( 
                           i 
                           ) 
                         
                       
                       , 
                       
                         b 
                         
                           ( 
                           i 
                           ) 
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
           
         
       
     
         [0031]    the sequential partition method is: 
         [0032]    i) firstly generating a set of K new vertices {b (1) , . . . , b (K) } based on the partition point r, wherein 
         [0000]        b   (i)   =b −( b   i   −r   i ) e   i   , i= 1, . . . , K  
 
         [0000]    wherein the subscript i indicates the i-th element of the vector and e i  is a vector with the i-th element being 1 and the others being 0; 
         [0033]    ii) sorting the K vertices as {b (i     1     ) , . . . , b (i     K     ) } in ascending order of the achievable weighted sum-rate, wherein i s  denotes an original index of the s-th vertex; 
         [0034]    ill) sequentially determining the corresponding vertices {a (i     1     ) , . . . , a (i     K     ) } as 
         [0000]    
       
         
           
             
               a 
               
                 ( 
                 
                   i 
                   s 
                 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         a 
                         , 
                       
                     
                     
                       
                         s 
                         = 
                         1 
                       
                     
                   
                   
                     
                       
                         
                           
                             a 
                             
                               ( 
                               
                                 i 
                                 
                                   s 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   r 
                                   
                                     i 
                                     
                                       s 
                                       - 
                                       1 
                                     
                                   
                                 
                                 - 
                                 
                                   a 
                                   
                                     i 
                                     
                                       s 
                                       - 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               e 
                               
                                 i 
                                 
                                   s 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                         , 
                       
                     
                     
                       
                         s 
                         &gt; 
                         1 
                       
                     
                   
                 
                 ; 
               
             
           
         
       
     
         [0035]    Step 8; calculating f(b) for each box [a,b]; 
         [0036]    wherein if f(b)&gt;f min , the associated vertex a is relocated as 
         [0000]    
       
         
           
             
               
                 a 
                 i 
               
               = 
               
                 
                   b 
                   i 
                 
                 - 
                 
                   
                     min 
                      
                     
                       ( 
                       
                         
                           
                             
                               f 
                                
                               
                                 ( 
                                 b 
                                 ) 
                               
                             
                             - 
                             
                               f 
                               min 
                             
                           
                           
                             
                               α 
                               i 
                             
                              
                             
                               ( 
                               
                                 
                                   b 
                                   i 
                                 
                                 - 
                                 
                                   a 
                                   i 
                                 
                               
                               ) 
                             
                           
                         
                         , 
                         1 
                       
                       ) 
                     
                   
                   × 
                   
                     ( 
                     
                       
                         b 
                         i 
                       
                       - 
                       
                         a 
                         i 
                       
                     
                     ) 
                   
                 
               
             
             , 
             
               i 
               = 
               1 
             
             , 
             … 
              
             
                 
             
             , 
             K 
           
         
       
     
         [0000]    wherein, ã i  is an i-th element of the relocated vertex; 
         [0037]    if f(b)≦f min , the box [a,b] is removed from the box set, i.e. =\[a,b]; 
         [0038]    Step 9: resetting the upper bound f max  as f max =max [a,b] εf(b); and 
         [0039]    Step 10: checking a relative error of the upper and lower bound;
       wherein if (f max −f min )/f min &gt;η, going back to the Step 5, otherwise, returning f max  and r min          
 
         [0041]    The monotonic optimization method proposed in the present invention adopts a sensible search scheme, a sequential partition method and a vertex relocation method to reduce the number of checking feasibility, therefore the computational complexity is decreased and the convergence is accelerated. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0042]      FIG. 1  is an illustration of an iteration for K=3; 
           [0043]      FIG. 2  is a diagram of comparison of convergence performance of the embodiment of the present invention, the conventional outer polyblock approximation method and branch-reduce-and-bound method in the scenario wherein M=3, K m =1, P m =5, T m =4; 
           [0044]      FIG. 3  is a diagram of comparison of convergence performance of the embodiment of the present invention, the conventional outer polyblock approximation method and branch-reduce-and-bound method in the scenario wherein M=2, K m =2, P=10, T m =4 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
       [0045]    The details of the embodiment of the present invention are specified below. The embodiment is based on the technical content of the present invention. The specific implementing process is presented. The scope of the present invention is not limited to the following embodiment. 
         [0046]    A weighted sum-rate maximization problem is 
         [0000]    
       
         
           
             
               maximize 
                
               
                   
               
                
               
                 
                   ∑ 
                   
                     m 
                     , 
                     k 
                   
                 
                  
                 
                   
                     α 
                     
                       m 
                       k 
                     
                   
                    
                   
                     R 
                     
                       m 
                       k 
                     
                   
                 
               
             
             = 
             
               
                 ∑ 
                 
                   m 
                   , 
                   k 
                 
               
                
               
                 
                   α 
                   
                     m 
                     k 
                   
                 
                  
                 
                   
                     log 
                     2 
                   
                   ( 
                   
                     1 
                     + 
                     
                       
                         
                            
                           
                             
                               h 
                               
                                 
                                   m 
                                   k 
                                 
                                 , 
                                 m 
                               
                             
                              
                             
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                         2 
                       
                       
                         
                           
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                                 ) 
                               
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                                
                             
                             2 
                           
                         
                         + 
                         
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                             m 
                             k 
                           
                           2 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   subject 
                    
                   
                       
                   
                    
                   to 
                    
                   
                       
                   
                    
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       
                         K 
                         m 
                       
                     
                      
                     
                       
                          
                         
                           w 
                           
                             m 
                             k 
                           
                         
                          
                       
                       2 
                     
                   
                 
                 ≤ 
                 
                   P 
                   m 
                 
               
               , 
               
                 ∀ 
                 m 
               
             
           
         
       
     
         [0000]    wherein K m  is a number of users in an m-th cell, T m  is an antenna number of an m-th base station, P m  denotes a maximum transmission power of the m-th base station, h m     k     ,n , is channel vector from a n-th base station to an m k -th user, σ m     k     2  represents a variance of a zero-mean complex Gaussian additive noise at the m k -th user and α m     k    is a weight of the m k -th user, wherein m,n=1, . . . , M, k=1, . . . , K m ; 
         [0047]    The monotonic optimization problem is 
         [0000]    
       
         
           
             
               maximize 
                
               
                   
               
                
               
                 f 
                  
                 
                   ( 
                   r 
                   ) 
                 
               
             
             := 
             
               
                 ∑ 
                 
                   m 
                   , 
                   k 
                 
               
                
               
                 
                   α 
                   
                     m 
                     k 
                   
                 
                  
                 
                   R 
                   
                     m 
                     k 
                   
                 
               
             
           
         
       
       
         
           
             
               subject 
                
               
                   
               
                
               to 
                
               
                   
               
                
               r 
             
             ∈ 
             ℜ 
           
         
       
     
         [0000]    wherein 
         [0000]    
       
         
           
             r 
             = 
             
               
                 ( 
                 
                   
                     R 
                     
                       1 
                       1 
                     
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     R 
                     
                       1 
                       
                         K 
                         1 
                       
                     
                   
                   , 
                   … 
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                   , 
                   
                     R 
                     
                       M 
                       1 
                     
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     R 
                     
                       M 
                       
                         K 
                         M 
                       
                     
                   
                 
                 ) 
               
               ∈ 
               
                 ℝ 
                 + 
                 K 
               
             
           
         
       
     
         [0000]    is the achievable rate vector and           denotes the achievable rate region, i.e. 
         [0000]    
       
         
           
             ℜ 
             = 
             
               
                 ⋃ 
                 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         
                           K 
                           m 
                         
                       
                        
                       
                         
                            
                           
                             w 
                             
                               m 
                               k 
                             
                           
                            
                         
                         2 
                       
                     
                     ≤ 
                     
                       P 
                       m 
                     
                   
                   , 
                   
                     ∀ 
                     m 
                   
                 
               
                
               
                 { 
                 
                   
                     
                       ( 
                       
                         
                           r 
                           
                             1 
                             1 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           r 
                           
                             M 
                             
                               K 
                               M 
                             
                           
                         
                       
                       ) 
                     
                     | 
                     
                       0 
                       ≤ 
                       
                         r 
                         
                           m 
                           k 
                         
                       
                       ≤ 
                       
                         R 
                         
                           m 
                           k 
                         
                       
                     
                   
                   , 
                   
                     ∀ 
                     m 
                   
                   , 
                   k 
                 
                 } 
               
             
           
         
       
     
         [0048]    The present invention includes the following steps of: 
         [0049]    Step 1: setting system parameters comprising: a cell/base station number M, the user number in the m-th cell K m , the antenna number of the m-th base station T m , the maximum transmission power of the m-th base station P m , wherein m=1, . . . , M, a 1 by T n  channel vector from the n-th base station to the m k -th user (i.e. the k-th user in the m-th cell) h m     k     ,n , wherein m,n=1, . . . , M, k=1, . . . , K m , the variance of the zero-mean complex Gaussian additive noise at the m k -th user σ m     k     2 , the weight of the m k -th user α m     k   , wherein m=1, . . . , M, k=1, . . . , K m ; 
         [0050]    In this embodiment, two scenarios are considered: M=3, K m =1, P m =5, T m =4 and M=2, K m =2, P m =10, T m =4, wherein m=1, . . . , M. 
         [0051]    In this embodiment, every entry of h m     k     ,n  is a zero-mean unit-variance complex Gaussian random variable, σ m     k     2 =1, α m     k   =1, wherein m,n=1, . . . , M, k=1, . . . , K m . 
         [0052]    Step 2: defining R m     k    as a rate of the m k -th user, 
         [0000]    
       
         
           
             
               R 
               
                 m 
                 k 
               
             
             = 
             
               
                 log 
                 2 
               
               ( 
               
                 1 
                 + 
                 
                   
                     
                        
                       
                         
                           h 
                           
                             
                               m 
                               k 
                             
                             , 
                             m 
                           
                         
                          
                         
                           w 
                           
                             m 
                             k 
                           
                         
                       
                        
                     
                     2 
                   
                   
                     
                       
                         ∑ 
                         
                           
                             ( 
                             
                               n 
                               , 
                               j 
                             
                             ) 
                           
                           ≠ 
                           
                             ( 
                             
                               m 
                               , 
                               k 
                             
                             ) 
                           
                         
                       
                        
                       
                         
                            
                           
                             
                               h 
                               
                                 
                                   m 
                                   k 
                                 
                                 , 
                                 n 
                               
                             
                              
                             
                               w 
                               
                                 n 
                                 j 
                               
                             
                           
                            
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         m 
                         k 
                       
                       2 
                     
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    wherein W m     k    is the T m  by 1 beamformer for the m k -th user, wherein m=1, . . . , M, k=1, . . . , K m ; 
         [0053]    formulating a achievable rate vector 
         [0000]    
       
         
           
             r 
             = 
             
               
                 ( 
                 
                   
                     R 
                     
                       1 
                       1 
                     
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     R 
                     
                       1 
                       
                         K 
                         1 
                       
                     
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     R 
                     
                       M 
                       1 
                     
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     R 
                     
                       M 
                       
                         K 
                         M 
                       
                     
                   
                 
                 ) 
               
               ∈ 
               
                 ℝ 
                 + 
                 K 
               
             
           
         
       
     
         [0000]    as well as a weighted sum-rate function 
         [0000]    
       
         
           
             
               
                 f 
                  
                 
                   ( 
                   r 
                   ) 
                 
               
               = 
               
                 
                   ∑ 
                   
                     m 
                     , 
                     k 
                   
                 
                  
                 
                   
                     α 
                     
                       m 
                       k 
                     
                   
                    
                   
                     R 
                     
                       m 
                       k 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    wherein ε           +   K  indicates that r is a positive real vector of a length K with K=Σ m=1   M K m ; 
         [0054]    Step 3: defining a box [a,b]={xε           +   K |a≦x≦b}; initializing the set of boxes as ={[0,b 0 ]}; assuming K 0 =0, a (Σ i=0   m−1 K i +k)-th element (associated with the m k -th user) of b 0  is log 2 (1+P m ∥h m     k     ,m   H ∥ 2 /σ m     k     2 ), wherein m=1, . . . , k=1, . . . , K m ; setting a termination accuracy η and a line search accuracy δ; 
         [0055]    In this embodiment, the accuracy parameters are set as η=0.01, δ=0.01. 
         [0056]    Since the iterations of the embodiment and the conventional methods are different, it is not meaningful to compare the number of iterations. As the main complexity derives from checking the feasibility of a point, the convergence performance is evaluated as a function of the feasibility checking times. The convergence performance is presented via the relative errors of the upper and lower bound with the optimal value, i.e. 
         [0000]      ( f   max   −f   opt )/ f   opt ,( f   min   −f   opt )/ f   opt    
         [0000]    wherein f opt  is the obtained optimal value of the weighted sum-rate; 
         [0057]    Step 4: initializing an upper bound f max  and a lower bound f min  of the weighted sum-rate, wherein 
         [0000]        f   max   =f ( b   0 ), f   min =max(α· b   0 )
 
         [0000]    wherein α is a weight vector comprising weights for all the users, i.e. 
         [0000]    
       
         
           
             
               α 
               = 
               
                 
                   ( 
                   
                     
                       α 
                       
                         1 
                         1 
                       
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       α 
                       
                         1 
                         
                           K 
                           1 
                         
                       
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       α 
                       
                         M 
                         1 
                       
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       α 
                       
                         M 
                         
                           K 
                           M 
                         
                       
                     
                   
                   ) 
                 
                 ∈ 
                 
                   ℝ 
                   + 
                   K 
                 
               
             
             ; 
           
         
       
     
         [0058]    Step 5: choosing a box [a,b] from that satisfies f(b)=f max , and then checking feasibility of a: whether a locates in a achievable rate region or not; wherein the feasibility is determined via a problem φ(a); 
         [0059]    the specified problem φ(a) is:
       maximize 0   subject to √{square root over (β m     k   )}∥A m     k   x+n m     k   ∥≦√{square root over (1+β m     k   )}(h m     k     ,m S m     k   x),∀m,k   p T x=0   ∥G m x∥≦√{square root over (P m )},∀m
 
with the following notations
       
 
         [0000]    
       
         
           
             
               
                 x 
                 m 
               
               = 
               
                 
                   [ 
                   
                     
                       w 
                       
                         m 
                         1 
                       
                       H 
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       w 
                       
                         m 
                         
                           K 
                           m 
                         
                       
                       H 
                     
                   
                   ] 
                 
                 H 
               
             
             , 
             
               m 
               = 
               1 
             
             , 
             … 
              
             
                 
             
             , 
             M 
             , 
             
               x 
               = 
               
                 
                   [ 
                   
                     
                       x 
                       1 
                       H 
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       x 
                       M 
                       H 
                     
                     , 
                     0 
                   
                   ] 
                 
                 H 
               
             
             , 
             
               
                 n 
                 
                   m 
                   k 
                 
               
               = 
               
                 
                   [ 
                   
                     0 
                     , 
                     
                       … 
                        
                       
                           
                       
                        
                       0 
                     
                     , 
                     
                       σ 
                       
                         m 
                         k 
                       
                     
                   
                   ] 
                 
                 T 
               
             
             , 
             
               
 
             
              
             
               
                 S 
                 
                   m 
                   k 
                 
               
               = 
               
                 [ 
                 
                   0 
                    
                   
                       
                   
                    
                   0 
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   
                     I 
                     
                       T 
                       m 
                     
                   
                    
                   
                       
                   
                    
                   …0 
                 
                 ] 
               
             
             , 
             
               
                 A 
                 
                   m 
                   k 
                 
               
               = 
               
                 diag 
                  
                 
                   ( 
                   
                     
                       
                         
                           h 
                           
                             
                               m 
                               k 
                             
                             , 
                             1 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           h 
                           
                             
                               m 
                               k 
                             
                             , 
                             1 
                           
                         
                       
                       
                          
                         
                           K 
                           1 
                         
                       
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     
                       
                         
                           h 
                           
                             
                               m 
                               k 
                             
                             , 
                             M 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           h 
                           
                             
                               m 
                               k 
                             
                             , 
                             M 
                           
                         
                       
                       
                          
                         
                           K 
                           M 
                         
                       
                     
                     , 
                     0 
                   
                   ) 
                 
               
             
             , 
             
               
 
             
              
             
                 
             
              
             
               
                 G 
                 
                   m 
                   1 
                 
               
               = 
               
                 
                   [ 
                   
                     
                       S 
                       
                         m 
                         1 
                       
                       H 
                     
                      
                     
                         
                     
                      
                     … 
                      
                     
                         
                     
                      
                     
                       S 
                       
                         m 
                         
                           K 
                           m 
                         
                       
                       H 
                     
                   
                   ] 
                 
                 H 
               
             
             , 
             
               
                 β 
                 
                   m 
                   k 
                 
               
               = 
               
                 
                   2 
                   
                     a 
                     
                       m 
                       k 
                     
                   
                 
                 - 
                 1 
               
             
             , 
             
               p 
               = 
               
                 
                   [ 
                   
                     0 
                     , 
                     
                       … 
                        
                       
                           
                       
                        
                       0 
                     
                     , 
                     1 
                   
                   ] 
                 
                 T 
               
             
           
         
       
     
         [0000]    wherein σ m     k    is a standard deviation of the zero-mean complex Gaussian additive noise at the m k -th user, I T     m    denotes a identity matrix with dimension T m ; 
         [0064]    Step 6: if the problem φ(a) is feasible, conducting a sensible search scheme for the box [a,b] to obtain a partition point r; 
         [0065]    wherein if the problem φ(a) is infeasible, updating the box set as =\[a,b] and calculating the upper bound f max =max [a,b] εf(b), then going back to the Step 5; 
         [0066]    the sensible search scheme is: 
         [0067]    denoting l ab  as the line connecting a and b; finding the intersection point c on the hyperplane {r|f(r)=f min } with the line l ab , i.e. 
         [0000]    
       
         
           
             c 
             = 
             
               a 
               + 
               
                 
                   ( 
                   
                     b 
                     - 
                     a 
                   
                   ) 
                 
                 × 
                 
                   
                     
                       f 
                       min 
                     
                     - 
                     
                       f 
                        
                       
                         ( 
                         a 
                         ) 
                       
                     
                   
                   
                     f 
                      
                     
                       ( 
                       
                         b 
                         - 
                         a 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
         [0068]    checking feasibility of c via the problem φ(a) in the Step 5 with a=c; 
         [0069]    wherein if φ(c) is feasible, a bisection line search is conducted along the line l cb  to obtain an intersection point on a Pareto-boundary; given a line search accuracy δ, two points r min  and r max  are acquired; the partition point is set as r=r max  while the lower bound is updated as f min =f(r min ); 
         [0070]    if φ(c) is infeasible, just set the partition point as r=c; 
         [0071]    Step 7: based on the partition point r, dividing the box [a,b] into K new boxes [a (i) ,b (i) ], i=1, . . . , K using a sequential partition method; then updating the boxes set as 
         [0000]    
       
         
           
             = 
             
               
                 \[ 
                  
                 
                   a 
                   , 
                   b 
                 
                 ] 
               
               ⋃ 
               
                 { 
                 
                   
                     ⋃ 
                     
                       
                         i 
                         = 
                         1 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       K 
                     
                   
                    
                   
                     [ 
                     
                       
                         a 
                         
                           ( 
                           i 
                           ) 
                         
                       
                       , 
                       
                         b 
                         
                           ( 
                           i 
                           ) 
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
           
         
       
     
         [0072]    the sequential partition method is: 
         [0073]    i) firstly generating a set of K new vertices {b (1) , . . . ,b (K) } based on the partition point r, wherein 
         [0000]        b   (i)   =b −( b   i   −r   i ) e   i   , i= 1, . . . , K  
 
         [0000]    wherein the subscript i indicates the i-th element of the vector and e i  is a vector with the i-th element being 1 and the others being 0; 
         [0074]    ii) sorting the K vertices as {b (i     1     ) , . . . , b (i     K     ) } in ascending order of the achievable weighted sum-rate, wherein i s  denotes an original index of the s-th vertex; 
         [0075]    iii) sequentially determining the corresponding vertices {a (i     1     ) , . . . , a (i     K     ) } as 
         [0000]    
       
         
           
             
               a 
               
                 ( 
                 
                   i 
                   s 
                 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         a 
                         , 
                       
                     
                     
                       
                         s 
                         = 
                         1 
                       
                     
                   
                   
                     
                       
                         
                           
                             a 
                             
                               ( 
                               
                                 i 
                                 
                                   s 
                                   - 
                                   1 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   r 
                                   
                                     i 
                                     
                                       s 
                                       - 
                                       1 
                                     
                                   
                                 
                                 - 
                                 
                                   a 
                                   
                                     i 
                                     
                                       s 
                                       - 
                                       1 
                                     
                                   
                                 
                               
                               ) 
                             
                              
                             
                               e 
                               
                                 i 
                                 
                                   s 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                         , 
                       
                     
                     
                       
                         s 
                         &gt; 
                         1 
                       
                     
                   
                 
                 ; 
               
             
           
         
       
     
         [0076]    Step 8; calculating f(b) for each box [a,b]; 
         [0077]    wherein if f(b)&gt;f min , the associated vertex a is relocated as 
         [0000]    
       
         
           
             
               
                 
                   a 
                   ~ 
                 
                 i 
               
               = 
               
                 
                   b 
                   i 
                 
                 - 
                 
                   
                     min 
                      
                     
                       ( 
                       
                         
                           
                             
                               f 
                                
                               
                                 ( 
                                 b 
                                 ) 
                               
                             
                             - 
                             
                               f 
                               min 
                             
                           
                           
                             
                               α 
                               i 
                             
                              
                             
                               ( 
                               
                                 
                                   b 
                                   i 
                                 
                                 - 
                                 
                                   a 
                                   i 
                                 
                               
                               ) 
                             
                           
                         
                         , 
                         1 
                       
                       ) 
                     
                   
                   × 
                   
                     ( 
                     
                       
                         b 
                         i 
                       
                       - 
                       
                         a 
                         i 
                       
                     
                     ) 
                   
                 
               
             
             , 
             
               i 
               = 
               1 
             
             , 
             … 
              
             
                 
             
             , 
             K 
           
         
       
     
         [0000]    wherein ã i  is an i-th element of the relocated vertex; 
         [0078]    if f(b)≦f min  the box [a,b] is removed from the box set, i.e. =\[a,b]; 
         [0079]    Step 9: resetting the upper bound f max  as f max =max [a,b]ε f(b); and Step 10: checking a relative error of the upper and lower bound; 
         [0080]    wherein if (f max −f min )/f min &gt;η, going back to the Step 5, otherwise, returning f min , f max  and r min . 
         [0081]      FIG. 2  compares the convergence performance of the embodiment of the present invention, the conventional outer polyblock approximation method (refer to Polyblock in figure) and branch-reduce-and-bound method (refer to BRB in figure) in the scenario wherein M=3, K m =1, P m =5, T m =4; 
         [0082]      FIG. 3  compares the convergence performance of the embodiment of the present invention, the conventional outer polyblock approximation method (refer to Polyblock in figure) and branch-reduce-and-bound method (refer to BRB in figure) in the scenario wherein M=2, K m =2, P m =10, T m =4. 
         [0083]    From  FIGS. 2 and 3 , it is observed that the embodiment of the present invention converges and achieves the global optimal solution faster compared with the conventional methods. 
         [0084]    While only selected embodiment has been chosen to illustrate the present invention, it will be apparent to those skilled in the art from this disclosure that various changes and modifications can be made herein without departing from the scope of the present invention.