English

Lagrange Multipliers

UniversityMathematicsQuadratic objective on a regular elliptical constraint with real-root stationary isolation including repeated roots

Unlock this lab with a pass

A pass opens every simulation on the site, including Present mode for class. The teaching notes below stay free for everyone.

See passesI already have a pass

One-time payment, no subscription.

About this experiment

Follow an objective around an ellipse and compare its gradient with a scaled constraint gradient. Find global constrained extrema and test degenerate cases in which every constraint point is stationary.

Learning goals

  • Interpret the regular constraint and its normal gradient.
  • Test the Lagrange multiplier stationarity equation.
  • Compare all stationary candidates before identifying global extrema.

How to use it

  1. 1Move the ellipse parameter and record objective and stationarity residual.
  2. 2Find the constrained minimum and inspect the coincident domain gradient arrows.
  3. 3Compare with the constrained maximum and its multiplier.
  4. 4Use a linear objective, then make the objective constant on a circle to inspect a fully stationary constraint.

The model

Both ellipse radii are positive, so its constraint gradient never vanishes.
The parameterisation makes constraint satisfaction exact.
A tangent-half-angle polynomial isolates real stationary candidates; repeated roots and the angle pi are included.

Assumptions

  • The objective is a general quadratic plus linear terms on a compact ellipse. The reported global extrema are selected from its stationary candidates.
  • Root isolation and stationarity checks use floating-point tolerances. A constant objective along the ellipse is detected separately.
  • The two arrows display grad f and lambda grad g in the domain. Their lengths and surface height are scaled for clarity; use numerical residuals for equality.

Try this in class

  1. Q1Why is a constrained maximum not necessarily a critical point of the unconstrained function?
  2. Q2Why does finding one stationary candidate not establish the global maximum?

Frequently asked questions

Can a zero objective gradient satisfy the multiplier equation?

Yes. On this regular constraint it satisfies the equation with multiplier zero. Nonzero parallel arrows are not required.

What happens if the objective is constant along the ellipse?

Every point is stationary, the global minimum equals the maximum and a single displayed representative does not mean the other points are excluded.