Antenna Beamforming Vector convex Optimization

1

wide_optimization

I have tried to solve this problem using this code-

cvx_begin
** variable u(N,1) complex**
** minimize(max(abs(conj(u’)a)))*
** subject to**
** conj(u’)at == 1 ;*
** conj(u’)aj == 0.01 ;*
cvx_end

But not getting the required results

2

Your formulation does not look even close to the image. It seems to correspond to the optimization problem at How can i solve the min-max convex optimization question . which is a different problem.

I can’t tell you what the correct formulations is, for the problem in the image (in your question in this thread) or even whether the problem is convex, because I have no idea what a(theta) or G_d(theta)` are. Do you know what they are? Have you proven the problem is convex? The convexity of the objective looks rather dubious.

3

This is my complete -

theta_radar_i = -10;
theta_radar_f = 10;
theta_increment = 1;
theta_radar = theta_radar_i:theta_increment:theta_radar_f;

LB = 2;
K = 2^(LB);
theta_cooms1 = -90:theta_increment:theta_radar_i-theta_increment;
theta_cooms2 = theta_radar_f+theta_increment:theta_increment:90;
theta_comms = [theta_cooms1 theta_cooms2];

theta = -90:theta_increment:90;
theta0_j = -50;

N = 10; %number of array elements
c = 3e8;
fc = 2e9;
lambda = c/fc;
d = lambda/2;

delta1 = [0.01 0.0033 0.0066 10e-4].^(1/2);
epsilon = sqrt(0.01);

n = 0:N-1;
a = exp(-1i2pid/lambda.n’.sind(theta_radar));
theta_t = 0;
theta_j = -40;
at = exp(-1i2pid/lambda.nsind(theta_t)).’;
aj = exp(-1i2pid/lambda.nsind(theta_j)).’;
ap = exp(-1i2pid/lambda.*n’.*sind(theta_comms));

Gd = exp(-1ipisind(theta_radar));

cvx_begin
variable u(N) complex
minimize((max(abs(Gd-conj(u’)*a))))
subject to
abs(conj(u’)ap) <= epsilonones(length(theta_comms),1)’ ;
conj(u’)*aj == delta1(3);
cvx_end

I have taken from a paper, In that paper it is clearly written that it is a convex problem.

4

it looks like you are using a toolbox function to prepare input data.

In any event, apparently theta is just a discrete set of values, and not a continuous optimization variable. Therefore, the problem is convex, and the CVX code is straightforward. I can’t speak as to the “correctness” of your input data.

5

I want to solve this problem using Interior Point Method. So is it possible to change the solver from default to Interior point method in CVX Toolbox?

6

All the solvers which can be called from CVX 2.2 use an interior point method, of one type or another. You can use cvx_solver to change the solver or change the default solver.