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6

The whole point is that you do NOT include the constraint $U = uu^T$ . That constraint is non-convex. Instead you include the constraint that $Z_j$ is (positive) semi-definite. This constraint is a convex relaxation of $U = uu^T$, obtained by Schur complement. I.e., the relaxed constraint, which is convex, is used instead of the original non-convex ...

5

Because the matrix KK is not PSD. It has minimum eigenvalue of -1. It is indefinite. Perhaps you are under the misapprehension that a symmetric matrix with all elements positive is PSD. As this example shows, that is not the case.

2

You can solve Question A as a second-order cone program like so: #!/usr/bin/env python3 import cvxpy as cp import numpy as np ########################## #Question A ########################## n = 50 #Arbitrary number of dimensions r = 10 #Arbitrary radius e = 3 #Epsilon value a =...

2

You are doing a broadcast (A*x), rather than matrix multiplication (A@x), so your code should look like this: #!/usr/bin/env python3 import numpy as np import cvxpy as cp A=np.array([[1,0,0],[0,1,0], [0,0,1]]) y=np.array([1,1,1]) # Upper bound for the constraint term upper=1 # Solve the optimization problem using CVXPY x = cp.Variable(3) ...

1

CVXPY's norm atom won't accept a raw Python list as an argument; you need to pass it a CVXPY expression. Stack the list of scalars into a vector using the hstack atom, like so: constraints = [cp.norm( cp.hstack([ y_hat[col] - cp.trace( np.transpose((B_hat_star[:,col][:,np.newaxis]*np.sqrt(L)*C_hat[col,:])) @ X) for col in ...

1

cvxpy's rules for disciplined convex programming are listed here. Notably, it states that: The DCP rules require that the problem objective have one of two forms: Minimize(convex) Maximize(concave) Indeed, your program runs if we remove the quad_form and doesn't run if we leave only the quad_form: prob = cp.Problem(cp.Maximize(cp.quad_form(x, ...

1

EDIT: Intermediate solution from cvxpy import * import numpy as np np.random.seed(1) n = 10 Sigma = np.random.randn(n, n) Sigma = Sigma.T.dot(Sigma) w = Variable(n) mu = np.abs(np.random.randn(n, 1)) ret = mu.T*w risk = quad_form(w, Sigma) orig_weight = [0.15,0.2,0.2,0.2,0.2,0.05,0.0,0.0,0.0,0.0] min_weight = [0.35,0.0,0.0,0.0,0.0,0,0.0,0,0.0,0.0] ...

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