Lagrange Multiplier Calculator
Optimize a quadratic objective with one linear constraint.
Table of Contents
How to Use
- Enter the quadratic coefficients a, b, c for f(x, y) = ax^2 + by^2 + cxy.
- Set the linear constraint d·x + e·y = k.
- Run the calculation to solve the Lagrange system for x, y, and λ.
- Review the optimized point and objective value on the constraint line.
How the Lagrange multiplier method works
Lagrange multipliers incorporate constraints into the objective by adding λ times the constraint equation. Stationary points satisfy the gradient equations and the constraint simultaneously.
- Form the Lagrangian L = f(x, y) + λ(d x + e y - k).
- Set partial derivatives ∂L/∂x, ∂L/∂y, and the constraint to zero.
- Solve the resulting linear system for x, y, and λ.
- Evaluate the objective at the stationary point to classify performance on the constraint.
Model assumptions in this calculator
- The objective is quadratic in x and y: ax^2 + by^2 + cxy.
- There is a single linear equality constraint d·x + e·y = k.
- A unique stationary point exists when the linear system is solvable.
- If the determinant is zero, the method cannot provide a unique solution.
Use this setup for quick intuition on constrained quadratics. More complex objectives or multiple constraints require a full symbolic solver.
Frequently Asked Questions
- What if d and e are both zero?
- The constraint becomes degenerate and cannot restrict the objective. Enter at least one non-zero coefficient for the constraint to find a valid stationary point.
- How do I know if this point is a minimum or maximum?
- This tool returns the stationary point on the constraint. To classify it, examine the quadratic form (e.g., via eigenvalues of the Hessian) or test nearby feasible points.
- Can I extend this to more variables?
- The method generalizes, but this calculator focuses on two variables with one linear constraint for clarity. For larger systems, apply the same gradient and constraint equations with additional multipliers.