At a Lagrange optimum ∇ f = λ ∇ g. Why must the two gradients be parallel there?

Because λ is chosen to make them parallel by definition

Because the constraint forces the two functions to have equal gradients

Because gradients of any two functions are parallel at a critical point

Physics I / Optimization: Critical Points and Lagrange Multipliers

Practice question · Multiple choice

At a Lagrange optimum $\nabla f = \lambda \nabla g$ . Why must the two gradients be parallel there?

Hints

Suppose the gradients were NOT parallel. Which way would you step?
You must stay on the constraint surface while stepping.

Show the answer

A. Because otherwise you could still improve along the constraint

Why

A component of $\nabla f$ along the surface is an unexploited improvement. Parallelism says there is none left, and $\lambda$ measures how much the optimum would improve if the constraint were relaxed.

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At a Lagrange optimum ∇f=λ∇g\nabla f = \lambda \nabla g∇f=λ∇g. Why must the two gradients be parallel there?

Practise Optimization: Critical Points and Lagrange Multipliers

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At a Lagrange optimum $\nabla f = \lambda \nabla g$ . Why must the two gradients be parallel there?