CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 51, 2016 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Tichun Wang, Hongyang Zhang, Lei Tian 
Copyright © 2016, AIDIC Servizi S.r.l.,  
ISBN 978-88-95608-43-3; ISSN 2283-9216 

Quadratically Constrained Quadratic Optimal Problem in 
Robust Relay Networks 

Hongyan Cai 
School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, No 10, 
Xitucheng Road, Haidian District, Beijing, China 
hy_cai75@163.com 

The paper investigates the problem of a distributed robust relay beamforming in one-way relay networks with 
imperfect channel state information. The aim is to maximize the SNR (signal-to-noise ratio) of the receiver with 
the constraint of the total transmits power of relays is less than a certain predefined threshold. Because the 
channel is imperfect and the objective function is fractional quadratic form, the optimal problem is a non-convex 
problem which is hard to be solved. We first change the fractional problem into a non-fractional one. Second, 
we consider the semi-definite programming (SDP) method efficiently by exploiting S-procedure theorem and 
relaxing the rank-one constrain to obtain the robust beamforming vector. The paper gives the schematic total 
algorithm too. Numerical results show that the proposed algorithm can solve the maximum problem with robust 
features efficiently and obtain satisfactory results. 

1. Introduction 

In this paper, we consider the problem of minimizing a fractional uncertainty quadratic function in the complex 
field as follows:  

1

2

( )
min

( )

. . ( ) 1.

f x

f x

s t g x

                   (1) 

Where ( ) , 1, 2H
i i i

f x x A x c i   , ic R , 
0

1

{ | 1}
k

j H

i i j i

j

A A u A uu


   , 
0
,

j n n

i i
A A


 are complex and 

Hermitian matrix. 2( ) || ||g x Bx , m nB C  , 
0

1

{ | 1}
k

j H

j

j

B B u B uu


   , the vectors

1 2
( , , , )

H

k
u u u u . Obviously, the disturbance vector u change in a certain set

{ | 1}.
k H

U u u u   ; and the disturbance matrix 
0

1

k
j

j
j

B B u B  is affine for u . 

The superscript “H” denotes the conjugate transpose. Furthermore, we require that the denominator of the 
objective function is nonzero, that is

2
| ( ) |f x N . Fn is feasible region of problem (1).  

Levinson studied first the complex optimization problem of linear optimization in complex space in 1966 
(Levinson, 1966). The linear fractional optimization in complex space was discussed by Swarup and Sharma in 
1970 (Swarup and Sharma, 1970). Later, more and more authors focus to the complex fractional optimization 
problems due to the wide range of applications in different areas including signal processing, wireless 
communications and so on (Ubaidulla et.al, 2007, Safavi et.al, 2012).  

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1651069

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Cai H.Y., 2016, Quadratically constrained quadratic optimal problem in robust relay networks, Chemical Engineering 
Transactions, 51, 409-414  DOI:10.3303/CET1651069   

409



2. A parametric schematic algorithm 

Because the object function of optimization problem (1) includes the disturbance vector u , which make the 
problem unsolvable . By introducing an auxiliary variable t , problem (1) can be translated into the following 
form.  

,

0

1 2
1

0 2

1

max

. . [ ] 1 ,

|| ( ) || 1,

1.

x t

k
H j

j
j

k
j

j
j

H

t

s t x C u C x c tc

B u B x

uu

               (2) 

Where, 
2 1

, 0,1, ,
j j j

C tA A j k . Let 0 1( ) ( , , , )kF x B x B x B x , so 

0 2

1

1 1
|| ( ) || ( ) ( )

H
k

j H

j
j

B u B x F x F x
u u

 1 1 0 1
1 0

0

H

H
uu

u I u

 

0 0 1

1

1
[ ] ( , , , )

H
k

H j H H H k

j
j

x C u C x x C x x C x x C x
u

 

Let 0 1( ) ( , , , )H H H kd x x C x x C x x C x , we have 

0 2

1 2 1 2
1

1 1
[ ] 1 ( ) ( ) (1 )

H
k

H j H

j
j

x C u C x c tc d x d x c tc
u u

 

Problem (2) can be rewritten as 

,

2

1 2

max

1 1(1 ) 0
. . ( ( ) ( ) ) 0,

0 0

1 1 0 1
( ( ) ( )) 0,

0 0

1 1 0 1
0.

0

x t

H

H

H

H

H

t

c tc
s t d x d x

u u

F x F x
u u

u I u

                  (3) 

To proceed, we will introduce a theorem S-procedure which can translate (3) into a certain SDP problem. 

Lemma 1 Let P and Q are both symmetric matrix, there is a vector 
0

u which satisfies 
0 0

( ) ( ) 0
T

u P u , then 

“ 0 0
T T

u Pu u Qu ” iff “ 0, s.t. Q P ”.(Ye, 2004; Boyd and Vandenberghe, 2004; Zhang 
et.al, 2011) 
We can translate the constraint conditions of the problem (3) into the semi-definite cone optimization problem. 
In fact, we let:  

1 0 1 0
, ( ) ( )

0 0 0

H
P Q F x F x

I

 

then the last two constraint conditions of problem (3) can be explained as:  
1 1 1 1

0 0

H H

P Q
u u u u

 

Let 
* 1

(1, 0, , 0) ,
T k

u then
* *

1 0
H

u Pu , by the lemma 1 we know: 
1

0,  such that 

1
Q P , namely 

 1
1 1

1

1 01 0 1 0
( ) ( ) ( ) ( ) 0, 0

00 0 0

H H
F x F x F x F x

II

 

In the same way, we have : 
2

0,  such that  

410



22

1 2 21 2

2

2

1 0 (1 ) 0(1 ) 0
( ) ( ) ( ) ( ) 0

0 00 0

H H
c tcc tc

d x d x d x d x
I I

 

For given t, problem (3) translates into a certain SDP problem as follow:  

,

2

1 2 2

2

1

1

1 2

max

(1 ) 0
. . ( ) ( ) 0 ,

0

1 0
( ) ( ) 0,

0

, 0.

x t

H

H

t

c tc
s t d x d x

I

F x F x
I

             (4) 

The major motivation for the problem (1) in this paper is from a practical issue in the communication system. 
Today, communications over wireless channels continue to be major challenges in technologies. More and 
more scholars study the collaborative use of amplify and forward (AF) or decode and forward (DF) protocols. 
Now most of the research works is based on the known ideal channel state information (CSI) which is hard to 
obtain in practice, however, non-ideal CSI will affect the performance of the network seriously [8]. So more and 
more scholars are interested in the robust beamforming algorithms in the non-cognitive or cognitive relay 
network and have published many papers about this topic (Cai and Wang, 2014; Veria et.al, 2008).  

3. Robust distributed beamforming problem  

We study a wireless network which includes two parts as Figure 1: a transmitter, a receiver, and N relay 
nodes .All nodes in network have been configured with one antenna. We assume that there is no direct link 
between the sources and the destinations, the help of relays is necessary to establish the communication link. 
Here we use two-step amplify-and-forward (AF) protocol. Assuming a flat fading scenario, denote 

i
f the 

channel coefficient from the transmitter to the i th relay and 
i

g represent the channel coefficient from the i th 

relay to the receiver.  

 

Figure 1: System model  

During the MAC, transmitter T transmits message s  to relays with the power
0

P  . Then the received signal at 

i th relay is given as follows:  

0 ii i R
r P f s n                    (5)

 

where, (0, )
iR R

n N N  represents the complex Gaussian noise at the i th relay. 

 During the BC, the retransmit signal iy of the i th relay is 

Relay 

1 

Relay 2 

 

Relay 

r 

Transmitter 

(T) 

Receiver 

(D) 

⋮ 

⋮ 

411



i i i
y w r                     (6) 

Where, iw is the complex beamforming weight. Then the received signals at receiver D can be written as:  

1

N

s i i d
i

y g y n                   (7) 

where, (0, )
d d

n N N represent the complex Gaussian noise at D. Substituting (5) and (6) into (7), we can 

obtain (8):  

0 0
1 1 1

signal, s noise,

= ( )
i i

T T

N N N

s i i i R d i i i i i R d
i i i

n

y w g P f s n n P w f g s w g n n

             (8) 

D
S , 

D
N are the received signal power and the noise power of D respectively, so we have 

2 2

0 0
1

2 2

1

{| | } {| | } ( )( )

{| | } {| | } ( )( )
i

N
H H

D T i i i
i

N
H H

P T i i R d R R d
i

S E s E P w f g s P g w f w f g

N E n E w g n n g w n w n g N

 

where,  

Let
1 2 1 2

( , , , ) , ( , , , )
H

N N
w w w w g g g g , 

1 2
( , , , )

N
f f f f , 

1
( , , )

NR R R
n n n . denotes 

the Hadamard product. 

The total transmitted power of relays is:  

2 2 2

1 1

{| | } | | {| | }
N N

H

T i i i

i i

P E y w E r w Dw
 

                     (9) 

Where 2 2 2
0 1 0 2 0

diag{ | | , | | , , | | }+
N R

D P f P f P f N I . 

In this paper, we assume the central processor is placed the inside of relays or the near of relays for 

searching CSI vector
i

f , 1i N . Because there is quantization error and feedback time-delay error in the 

actual transmission, ideal CSI vector , 1, ,
i

g i N can’t be obtained, CSI error model may be expressed 

as： 
,g g g  g  denotes the observed channel vector and the error vector g is bounded, given by

2

2
{ ||| || }

N
g a a

. 

The SNR of SD is defined as D
D

S
SINR

N
. 

In the section, our aim is to obtain the optimal robust beamforming weight coefficient and maximize the SNR of 
D.  

So the optimal model can write as following:  

0
( )( )

max
( )( )

. . ,

.

H H

H H

R R d

H

H

P g w f w f g

g w n w n g N

s t w Dw P

g g

               

(10) 

Because of ( )( )
H H H

a b a b aa bb and so the optimization (10) can be rewritten as: 

412



0
( )

max
( )

. . ,

.

H H H

H H H

R R d

H

H

P g ff ww g

g n n ww g N

s t w Dw P

g g

                                                                    (11) 

Let 
H

W ww , so rank( ) 1W ,and rewrite the optimization problem(11): 

0
( )

max
( )

. . ,

,

( ) 1.

H H

H H

R R d

H

P g ff W g

g n n W g N

s t D W P

g g

Rank W

                                                                      (12) 

Due to the rank-one constraint, the optimization problem (12) is hard to be solved. Therefore we resort to 
relaxing (13) by deleting the rank-one constraint ( ) 1Rank W , namely 

0
( )

max
( )

. . ,

,

H H

H H

R R d

H

P g ff W g

g n n W g N

s t D W P

g g

                                                                      (13) 

By using the method that introduced in the second section, we can obtain optimal solution *W of the 
optimization (13) by MATLAB. If the optimal solution *W of (13) is rank-one, the optimal solution 

*
w of (12) 

can be obtained by using eigenvalue decomposition. But if *W is not rank-one, we can use randomization and 
eigenvector approximation method to obtain the optimum *w of (12).(Huang and Zhang,2010) 

4. Numerical solution and analysis  

In this section, set
0

10P dB , relay number 5N . The channel coefficient Vectors f , g  and the CSI 

error g  can be generated following complex Gaussian distribution (0, )CN I . The complex Gaussian noise 

power of all nodes are 1, that is , , 1R D dN N N . 

In Figure 2, we illustrate the achieved SNR of D as the total transmit power of relays P , consider the channel 

error bounds 0.1 , 0.7 and the non-robust case. It is observed that SNR of D increase obviously 

with the total relay power enlarging no matter whether 0.1 , 0.7 or the non-robust case. It is obvious 
that the proposed robust algorithm outperforms the non-robust scheme.  

  

Figure 2: Output SNR maximum versus total relay power 

0 2 4 6 8 10 12 14 16 18 20
0

5

10

15

20

25

the total transmitted power of relays PR(dB)

S
N

R
 o

f 
th

e
 c

o
g
n
it
iv

e
 r

e
la

y
 n

e
tw

o
rk

 u
s
e
r 

 

 

=0.7

Nonrobust

=0.1

413

http://www.baidu.com/link?url=V8GWo8fc9NPXmUMFgbELqixw2rxPtZT0GL6U0HbpJoxWRAF1C4biZabkJjRRI__W1WuWc7tZGrd1kWulBCFHUk8gI2hmJnxYnwmd1RNx1ji


5. Conclusions 

In this paper, we proposed the robust relay beamforming for RN using semi-definite programming method for 
maximizing the SNR of D. The optimization problem is non-convex because of the channel information 
imperfect. We introduce a robust algorithm for this uncertain problem by exploiting S-procedure and relaxing 
the rank-one constrain. Numerical result shows that the proposed algorithm can solve the maximum problem 
with robust features efficiently and obtain satisfactory results too. This paper consider the channel uncertainty 
at the second process only, then future work will focus on channel errors in the two transmitted phases.  

Acknowledgements 

This work is supported by the National Science Foundation of china (NSFC) under grant 61372114 and 

61571054. 

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