Augmented Lagrangian One Step 1D

Problem

Minimize x^2 subject to x = 1 with the augmented Lagrangian. With multiplier lambda and penalty rho, the primal update is x_new = (lambda + rho) / (2 + rho) and the multiplier update is lambda_new = lambda + rho * (x_new - 1).

Given lambda and rho, return [x_new, lambda_new].

Hint

Minimize x^2 + lambda (x - 1) + (rho / 2) (x - 1)^2 over x.

Test cases

case 1

in: [0,1]

expected: [0.3333333333333333,-0.6666666666666667]

case 2

in: [2,4]

expected: [1,2]

case 3

in: [-1,10]

expected: [0.75,-3.5]

Starter codePython

def augmented_lagrangian_one_step_1d(multiplier, rho):
    # Your code here
    pass

3 test cases · run your code against them in the editor