import numpy as np
import cvxpy as cp


## PROBLEM 3
# Create data
m = 30
n = 10
np.random.seed(2021)
A = np.random.randn(m, n)
y = np.random.randn(m)

# Set up CVX problem
x = cp.Variable(n)
objective = cp.Minimize(0.5*cp.norm(A@x - y,2)**2)
constraints = [0 <= x]
prob = cp.Problem(objective, constraints)

# The optimal objective value is returned by `prob.solve()`
result = prob.solve()
# The optimal value for x is stored in `x.value`
xstar = x.value
# The optimal Lagrange multiplier for a constraint is stored in
# `constraint.dual_value`
lambdastar = constraints[0].dual_value
    

Q = A.T @ A
b = A.T @ y
print(np.linalg.norm((Q @ xstar - b) - lambdastar))