Let W = [\alpha, \beta]^\intercal be a variable matrix, where \beta>0 can be fixed as constant in some use-case and 0<\alpha<1. Now, after some transformation, we define Z = [W^\intercal 1]^\intercal and we further define Y= Z\,Z^\intercal.
Now I want to minimize \text{Tr}(Q\,Y) using CVX. Assume that Q is a positive real matrix of size (3\times 3). I guess, we can take variable Y(3,3)after begin_cvx.
- If I initialize \alpha with random value outside of begin_cvx, CVX does not take this as a variable, thus does not give any optimum values of \alpha, CVX only tries to find optimum values of Y. Based on the random value of \alpha, cvx gives optimum Y. (correct me if I am wrong)
Question-1: What is the right place to define \alpha, inside of CVX, using expression \alpha or outside?
Question-2: I declare \alpha=\text{rand}(1,1) inside of cvx_begin, and then again if I try to impose a constraint 0<Q_\alpha Y<1 to hold 0<\alpha<1, where Q_\alpha is the matrix of size (3\times 3) , then the optimum value gives NaN. See the below, (if I uncomment 1>=(Q_alpha*Y_k)>=0, then it gives NaN)
clear all
clc;
e2 = transpose([0 1 0 0 0 0]);
Q_alpha = [zeros(6,6) 0.5*e2;0.5*transpose(e2) 0];
e4 = transpose([0 0 0 1 0 0]);
Q_r = [zeros(6,6) 0.5*e4;0.5*transpose(e4) 0]
lambda_k_D_in =15000; % Total size in byte
zeta = 0.2
P_k = transpose([lambda_k_D_in, 0, 0, zeta, 0, 0])
Q_k = [zeros(6,6) 0.5*P_k;0.5*transpose(P_k) 0]
Q_r= [zeros(6,6) 0.5*e4;0.5*transpose(e4) 0]
M=3
r_k_i = 4*10^5;
cvx_begin
variable Y_k(7 7)
alpha =rand(1,1)
gamma = rand(1,1)
W_k = transpose([0.2 alpha gamma r_k_i 2 M])
Z_k = transpose([transpose(W_k) 1]);
Y_k==Z_k*transpose(Z_k);
minimize(trace(Q_k*Y_k))
trace(Q_r*Y_k)>=0
% 1>=(Q_alpha*Y_k)>=0
cvx_end
Y_k
trace(Q_k*Y_k)
Q_k