1.
Hermitian transpose or conjugate transpose:
- Hermitian transpose or conjugate transpose of an m-by-n matrix 'A' with complex entries is the n-bym matrix A∗ obtained from A by taking the transpose and then taking the complex conjugate of each
entry. (The complex conjugate of a + bi, where a and b are reals, is a − bi.)
If
-
2.
 2  i
1
A
1

i
i 

Example:
then
1 i 
1
AH  
 i 
 2  i
Trace (Linear algebra)
- The trace of an n-by-n square matrix A is defined to be the sum of the elements on the main
diagonal (the diagonal from the upper left to the lower right) of A, i.e.,
n
tr ( A)   aii  a11  a22  ...  ann .
i 1
-
Example:
 a11 a12 a13   1 0 3 

 

Let A is a matrix, with: A   a21 a22 a23   11 5 2 
 a a a   6 12  5 

 31 32 33  
3
Then: tr  A   aii  a11  a22  a33  1  5  (5)  1.
i 1
I.
Inequality transformation:
1. We have √𝑥𝑦 is a concave function wity x>0,y>0 in “A New Design Paradigm for Secure Full-Duplex
Multiuser Systems”
2. xy <= (x^2+y^2)/2. Proof: https://math.stackexchange.com/questions/357272/how-can-i-prove-that-xy-leqx2y2
1
1. 𝑙𝑜𝑔2 (1 + ) (checked) is a convex function as in paper: Yang Wu, Weiwei Yang, Xinrong Guan, and
𝑥𝑦
Qingqing Wu “Energy-Efficient Trajectory Design for UAV-Enabled Communication under Malicious
Jamming” under review IEEE CL, C:\Users\hieu.tran-dinh\OneDrive\PhD\Review\2020\ICL\1
Proof: http://theanalysisofdata.com/probability/C_4.html
http://control.ucsd.edu/mauricio/courses/mae280a/lecture11.pdf
https://www.cse.iitk.ac.in/users/rmittal/prev_course/s18/reports/7psdmatrices.pdf
2.
3.
4.
5.
6.
Fractional programming method: Dinkelbach’s algogithm (i.e., objective function is a fractional function).
Path-following algorithm in the paper: why by applying IA method, we can guarantee the solutions can
converge to at least a locally optimum solution.
https://ieeexplore.ieee.org/document/7296696/?fbclid=IwAR30at08FIjZMc79U9-4MMP6JJIuBjutHyg6VBYVrlYaCBgIpuRzkvZD6o
(x+y)^2 and (x-y)^2 are convex functions
𝑙𝑜𝑔2 (1 + 𝑥𝑦) w.r.t. x is a concave function as in Appendix A of paper “Hua, Meng, Luxi Yang,
Chunguo Li, Qingqing Wu, and A. Lee Swindlehurst. "Throughput Maximization for UAV-aided
Backscatter Communication Networks." IEEE Transactions on Communications 68, no. 2 (2019): 12541270.”
Proof Eq. 20 in “Spectral and Energy Efficiencies in Full-Duplex Wireless Information and Power
Transfer”
3.
4.
Find the lower bound for function log_2(1+x/y): Check it with [1] “Spectral and Energy Efficiencies in FullDuplex Wireless Information and Power Transfer”, wherether or not the function –ln(1-x/z) convex when
z>x. As in appendix A of [1], this is obviously that –ln(1-t) is convex and increasing in the domain 0 <=
t <1. Thus, –ln(1-x/z) convex when z>x.
4. Error with CVX when using log function: http://ask.cvxr.com/t/cvxquad-how-to-use-cvxquads-padeapproximant-instead-of-cvxs-unreliable-successive-approximation-for-gp-mode-log-exp-entr-rel-entr-kldiv-log-det-det-rootn-exponential-cone-cvxquads-quantum-matrix-entropy-matrix-log-relatedfunctions/5598
If there exists log function containing variables inside this, it is better to use Yalmilp.
Log-distance path loss model:
- This sytem model is used a lots in Satellite and UAV communciations whereas the channel between
Satellite/UAV to Gus is dominated by LoS channel. However, there exists 2 styles of presentations for
this model.
-
In [1], they model the path loss model as logarithm function and in Rui Zhang paper, they user nonlograrithm model. This can be clarified in https://en.wikipedia.org/wiki/Log-distance_path_loss_model
[1] X. Li, W. Feng, Y. Chen, C. Wang and N. Ge, "Maritime Coverage Enhancement Using UAVs Coordinated
With Hybrid Satellite-Terrestrial Networks," in IEEE Transactions on Communications, vol. 68, no. 4, pp. 23552369, April 2020, doi: 10.1109/TCOMM.2020.2966715.
5.
Semi definite matrix:
https://www.cse.iitk.ac.in/users/rmittal/prev_course/s14/notes/lec11.pdf
6. Composition with an affine mapping: How to proof a function is convex
- Section 3.2.2, page 93/720 Convex Optimization book
- http://www.princeton.edu/~aaa/Public/Teaching/ORF363_COS323/F14/ORF363_COS323_F14_
Lec6.pdf
- http://web.mit.edu/~jadbabai/www/EE605/lectures/functions.pdf
-
f(Ax + b) is convex if f is convex
Trace of a matrix is sum of eigenvalues, determinant of a matrix is product of eigenvalues.
If both trace and determinant is positive -> eigenvalues is positive -> Function is convex
https://en.wikipedia.org/wiki/Definite_matrix#Characterizations
http://www.math.udel.edu/~angell/Opt/conv_fcn.pdf
https://en.wikipedia.org/wiki/Definite_matrix
- Then f is convex if and only if dom f is convex and its Hessian
is positive semidefinite: Steven Boy Book 3.1.4 Second-order
conditions
-