The Method of Lagrange Multipliers
“grad(f) = lambda grad(g), collinear constraint normals, shadow prices, and constrained critical points.”
1.The Geometry of Equality Constrained Extrema
§ 4.1.Consider minimizing f(x) constrained to surface g(x) = 0.
Geometric Principle:
At the optimum, the contour curve of f must be tangent to the constraint curve g(x) = 0. If they crossed transversely, moving along the constraint curve would continue to increase or decrease f.
Since both curves share a tangent hyperplane at the contact point, their normal vectors (gradients) must be collinear:
grad(f(x)) = lambda grad(g(x*)).
The Lagrangian Formulation:
L(x, lambda) = f(x) - lambda * g(x).
Setting grad L = 0 converts the constrained problem into an unconstrained root-finding problem in n + 1 variables.
2. Worked Practice Problem: Multi-Constraint Lagrange Optimization
Set up the gradient proportionality system alongside both equality constraint equations.
Problem: Find the minimum value of f(x, y, z) = x^2 + 2*y^2 + z^2 subject to two simultaneous equality constraints: g1(x, y, z) = x + y + z - 1 = 0 and g2(x, y, z) = x - y = 0.
Step 1: Formulate Lagrangian with two multipliers lambda1 and lambda2:
L(x, y, z, lambda1, lambda2) = x^2 + 2y^2 + z^2 - lambda1(x + y + z - 1) - lambda2*(x - y).
Step 2: Compute partial derivatives:
dL/dx = 2x - lambda1 - lambda2 = 0 => 2x = lambda1 + lambda2
dL/dy = 4y - lambda1 + lambda2 = 0 => 4y = lambda1 - lambda2
dL/dz = 2z - lambda1 = 0 => 2z = lambda1 => z = lambda1 / 2.
Step 3: Use constraint g2: x - y = 0 => y = x:
Substitute y = x into gradient equations:
2*x = lambda1 + lambda2
4*x = lambda1 - lambda2
Add equations: 6x = 2lambda1 => x = lambda1 / 3.
Then y = x = lambda1 / 3.
Step 4: Substitute x, y, z in terms of lambda1 into constraint g1: x + y + z = 1:
(lambda1 / 3) + (lambda1 / 3) + (lambda1 / 2) = 1
(2lambda1 + 2lambda1 + 3lambda1) / 6 = 1 => 7lambda1 / 6 = 1 => lambda1 = 6 / 7.
Step 5: Find optimal coordinates (x, y, z*):
x* = (6/7) / 3 = 2 / 7.
y* = 2 / 7.
z* = (6/7) / 2 = 3 / 7.
Step 6: Compute minimum value f(x, y, z*):
f(2/7, 2/7, 3/7) = (2/7)^2 + 2*(2/7)^2 + (3/7)^2
f = (4 / 49) + 2*(4 / 49) + (9 / 49) = (4 + 8 + 9) / 49 = 21 / 49 = 3 / 7 ≈ 0.4286.
Conclusion: The constrained minimum occurs at (2/7, 2/7, 3/7) with minimum value 3/7.
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