Cvx maximization problem

1

Hi, I have an optimization problem whose optimum solution is known. However I could not re-produce it myself.

The problem

image
image.png789×397 58.1 KB

I know K (16x16) and s_i =SMP(:,ii) where SMP is (4x16) . Both of these variables are non-complex. W is the optimum precoding matrix. I wrote this problem as below

cvx_begin 

variable t nonnegative
variable W(4,4) nonnegative diagonal
maximize t
subject to 
t <= transpose(vec(W))*K*vec(W)
for ii=1:16
W*SMP(:,ii)>=zeros(4,1)
W*SMP(:,ii)<=ones(4,1)
end
cvx_end

but it didn’t work. Can you help me?

2

It appears that K should really be indexed by i and j, and that you need a constraint for every i \ne j, yet you only have a single constraint involving K, and you do not show us what that K is.

You declared W as diagonal, yet the image has W' as well as W, which would seem strange if W is diagonal (which isn’t mentioned in the image).

There may be more wrong as well, but perhaps get started on this.

3

This is norm maximization and does not look convex.

4

Thank you for your prompt response. H is a channel matrix 4x4 and its values are given below:

H= 1e-5 * [0.6250 0.6101 0.5958 0.6101;
0.6101 0.6250 0.6101 0.5958;
0.5958 0.6101 0.6250 0.6101;
0.6101 0.5958 0.6101 0.6250]

K is a matris with 16x16 and it is calculated by

K = transpose(kron(H,D.’))*kron(H,D.’);

where D is (s_i-s_j) where s is the constellation point and its dimension is 120x4.

image
48765FAD-2B4F-456A-923A-D4CB7E58CAD1.jpeg2026×447 234 KB

In the paper, it is stated that “The diagonal precoder can be obtained by solving the optimization problem in (12) with an additional constraint w_kl = 0 when k ̸= l.
”
The diagonal precoder
WD obtained by solving (12) is given by WD = diag{0.2154, 0.4290, 0.3568, 0.5}

I hope these info are useful to describe my problem.

5

Have you proven this is a convex optimization problem? Per the rules of the forum, a statement in a book or paper asserting convexity or solvability by CVX doesn’t count.

6

Thank you for your help. I try to prove it.

Best,
Suzy

7

sorry to bother you ,i wanna ask do you know how to maximize an convex expression?the matlab shows it can’t maximize an convex expression…

8

You use a non-convex solver. And that probably is the case for your other question, Disciplined convex programming error: Cannot perform the operation: {complex affine} .* {log-affine} . So not CVX. You might be able to use YALMIP.

The only non-convex problems which can be handled in CVX are those for which the only non-convexity is binary or integer constraints on the variables.