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Lagrange Multiplier Geometry

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About This MicroSim

This visualization demonstrates the geometric interpretation of Lagrange multipliers. The key insight is that at a constrained optimum, the gradient of the objective function must be parallel to the gradient of the constraint (the constraint normal).

Problem Setup

  • Objective: Maximize f(x,y) = x + 2y
  • Constraint: x² + y² = 4 (circle of radius 2)

How to Use

  1. Drag the Red Point: Move it along the constraint circle
  2. Watch the Arrows: Blue is ∇f (gradient of objective), Red is ∇h (constraint normal)
  3. Find Optimum: Click the button to animate to the optimal point
  4. Toggle Fields: Show gradient and normal fields to see the global picture

Key Insight

At the optimal point:

\[\nabla f(\mathbf{x}^*) = \lambda \nabla h(\mathbf{x}^*)\]

The gradients are parallel! This means you cannot improve f while staying on the constraint surface.

Understanding the Visualization

  • Gray Lines: Level curves of f(x,y) = x + 2y (higher = upper-right)
  • Bold Circle: The constraint h(x,y) = x² + y² - 4 = 0
  • Blue Arrow: Direction of steepest increase of f (∇f)
  • Red Arrow: Normal to the constraint (∇h)
  • Green Point: At optimum, both gradients are parallel

Lesson Plan

Learning Objectives

  • Understand the geometric meaning of Lagrange multipliers
  • Visualize why gradients must be parallel at constrained optima
  • Connect the algebraic condition ∇f = λ∇h to geometry

Suggested Activities

  1. Exploration: Drag the point around the entire circle and observe when gradients align
  2. Two Optima: Notice there's both a maximum and minimum on the circle
  3. Field Visualization: Enable gradient field to see why ∇f is constant for linear f
  4. Lambda Interpretation: Watch how λ changes as you move the point

References