Page 326 - Kaleidoscope Academic Conference Proceedings 2024

2024 ITU Kaleidoscope Academic Conference

optimization subproblem can be formulated as: s.t.(7) , (7) , (16) (18)b

We can observe that (18)b are all convex, and (18)a is the
∑︁

max (11)a difference between two concave functions, the optimization
{ }
=1 problem (18) is a typical DC programming problem. For
∈ K, we define function ( ) that satisfies ( ) = 0 as
s.t.(7) , (7) (11)b
,
We can observe that the optimization problem (11) satisfy ( ) = , ≥ + 1, (19)
ln2
linear programming. Therefore, we can directly derive CPU
frequency in closed-form expression. Based on constraints then the gradient of 2 at p can be expressed as
(11)b, the range of can be denoted by
∑︁

√︄ ∇ 2 (p) = , (20)
  Í
, + 1
 
  3 −  =1 = +1


0 ≤ ≤ min , . (12)

 
Then, by defining ( − 1)-th iteration feasible solution
 
 
( −1) (0)
p , and initializing a feasible solution p , the
Since the objective function (11)a monotonically increases ( )
∗
with , the optimal of local CPU frequency is the upper optimal solution p of -th iteration can be obtained
by converting problem (18) to
bound of , which is given by
( −1)
( −1) ( −1)
√︄ max 1 (p)− 2 (p )− ∇ 2 (p ), p − p (21)a
 
  p

∗   3 − 
= min , . (13)

  s.t.(18) (21)b
 
 
where ⟨., .⟩ is the operation of inner product. Obviously,
3.3 Optimize Transmission Power problem (21) is convex. Therefore, we can efficiently solved
it via utilizing CVX toolbox [24].
Thirdly, we focus on transmission power optimization. Given
W, { }, and Θ, the transmission power optimization 3.4 Optimize Phase Shift
subproblem can be rephrased as
Finally, we optimize the phase matrix Θ. Given W, { }
!
2
∑︁
|w h ( )| and p, the IRS phase shifts optimization subproblem can be
max log 2 1 + Í (14)a
p 2 2 reformulated as
=1 = +1 |w h ( )| +

s.t.(7) , (7) , (7) (14)b ∑︁
max log (1 + ) (22)a
2
Θ
Because objective function (14)a is nonconvex, problem (14) =1
|w h ( )| 2
is still difficult to handle. By defining , = 2 , the s.t.(7) , (7) (22)b

objective function (14)a is reexpressed as 2

To transform the terms |w (GΘh + h , )| into tractable

! !
form, we define v = [ 1 , · · · , ] , where = , ∀ ∈
∑︁
∑︁
∑︁
∑︁
ˆ
log 2 , + 1 − log 2 , + 1 (15) N. Then, we define h , = (w G) ◦ h , where ◦ denotes the

=1 = =1 = +1 operation of Hadamard product. Correspondingly, we can
obtain
Subsequently, by substituting the expression of into (7)d,
2 ˆ 2 (23)
|w GΘh | = |v h , | .
the constraint (7)d can be rephrased as
Besides, by defining , = w h , , the problem (22) can be

!

reformulated as
∑︁

, ≥ , + 1 , ∀ ∈ K. (16)

!
= ∑︁ ˆ 2
|v h , + , |
max log 2 (24)a
ˆ
Obviously, (16) is an affine constraint, that is, a convex v 1 + Í |v h , + , | + 2
2
=1 = +1
constraint. Correspondingly, we define functions
ˆ 2
s.t.|v h , + , |
!
∑︁
∑︁
!
1 (p) = log , + 1 , ∑︁ (24)b
2 ˆ 2 2
≥ |v h , + , | + , ∀ ∈ K
=1 =
(17)
∑︁ ! = +1
∑︁
2 (p) = log 2 , + 1 . | | = 1, ∀ ∈ N, (24)c
=1 = +1
Nevertheless, problem (24) is still non-convex. Then, by
Hence, the transmission power optimization subproblem can defining
be reformulated as " #
ˆ ˆ h
ˆ
h h , v
, , , , ¯ v = , (25)
max 1 (p) − 2 (p) (18)a Φ , = ˆ 1
p h , , , ,
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