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Interactive 3D/Lagrange Multipliers
∇f = λ∇g at the optimum - gradients are parallel
Target (a, b)
a = 1.50
-22
b = 1.00
-22
Constraint Radius r
r = 1.50
0.33.0
Find Optimum
Results
x*: -
y*: -
λ: -
f(x*,y*): -
Key Idea
At the optimum, ∇f = λ∇g - the gradients are parallel. λ (lambda) is the Lagrange multiplier: the rate of change of the optimal value with respect to the constraint.
Try: Place (a,b) inside the circle (r larger than distance). What happens? The constraint becomes inactive.

Lagrange Multipliers - Interactive Visualization

Lagrange multipliers solve constrained optimization: minimize f(x,y) subject to g(x,y) = 0. The key insight is that at the optimum, the gradients ∇f and ∇g must be parallel: ∇f = λ∇g. This visualization animates the search for where the objective's gradient aligns with the constraint's gradient - the Lagrange condition. This underlies SVM dual formulation and neural network pruning.

  • See objective contours and constraint curve on the same plot
  • Watch gradient arrows ∇f and ∇g become parallel at the optimum
  • Adjust objective center (a,b) and constraint radius r
  • Animate the search to the constrained optimum
  • Connect to SVM dual, KKT conditions, and variational problems

Part of the EngineersOfAI Interactive 3D - free interactive visualizations covering every major concept in machine learning and AI engineering. Hover any element for a plain-English explanation. No code required.