ECE720
Homework 5
1. “Hello World” in CVX.
Consider the following complex least e∞-norm problem:
with complex variables x ∈ C
n and complex parameters A ∈ C
m×n and b ∈ C
m. Note that
Generate the problem data as follows:
• m = 30 and n = 100
• A and b can be randomly generated according to Gaussian distribution. The Matlab command will be
A = randn(m,n) + i*randn(m,n); b = randn(m,1) + i*randn(m,1);
Note that Matlab defines variable i as i =
√
−1 by default.
(a) Use CVX to solve the problem with complex variables directly.
Hint: You can use norm(x, inf) with complex arguments directly in CVX. CVX rec-ognizes norm(x, inf) as a valid convex function. You need to declare variables to be complex in the variable statement.
(b) We can convert the problem into a SOCP with real variables as follows. First, define the optimization variable as
Then define
Finally, define Ci ∈ R
2×(2n) as a matrix with all zero elements except (Ci)1,i = 1 and (Ci)2,n+i = 1, such that
The equivalent SOCP will be
with optimization variables t ∈ R and z ∈ R
2n
.
Solve the above SOCP using CVX.
(c) Verify the optimal solutions and optimal values obtained in two approaches are the same.
(d) Please submit your solution in Blackboard together with your codes.