代写ECE720 Homework 5调试R程序

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.