MediumCalculusca-170

Optimum Sensitivity

Problem

Minimize f(x, y) = 0.5*a*x^2 + 0.5*b*y^2 subject to the equality constraint x + y = s.

The KKT solution is x* = s*b/(a + b) and y* = s*a/(a + b). Return the sensitivities [dx*/ds, dy*/ds], which are b/(a+b) and a/(a+b). The parameter s cancels, and a + b > 0.

Hint

Solve the equality-constrained problem, then differentiate the optimizer with respect to s.

Test cases

case 1

in: [1,1,5]

expected: [0.5,0.5]

case 2

in: [2,1,3]

expected: [0.3333333333333333,0.6666666666666666]

case 3

in: [3,6,1]

expected: [0.6666666666666666,0.3333333333333333]

case 4

in: [1,4,2]

expected: [0.8,0.2]

Starter codePython

def optimum_sensitivity(a, b, s):
    # Return [dx_ds, dy_ds]
    # Your code here
    pass

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