Type a command or search the notes. Tab cycles completions. Up and down recall history when blank. Escape dismisses, then leaves the prompt.
MATH
[1251]

Methods from Constrained extrema

All methods from Constrained extrema.

Back to Constrained extrema

Optimise subject to one equality constraint

To optimise ff subject to g=0g=0:

  1. check where ∇g=0\nabla g=\mathbf{0} on the constraint and retain any such singular feasible points for separate analysis;
  2. compute ∇f\nabla f and ∇g\nabla g;
  3. solve ∇f=λ∇g,g=0;\nabla f=\lambda\nabla g, \qquad g=0;
  4. evaluate ff at every solution;
  5. compare the values to identify the constrained maximum and minimum.

The system has three unknowns xx, yy and λ\lambda in the two-variable case. Symmetry and factorisation often reduce the algebra. Solving the multiplier equations produces candidates; it does not replace checking singular points or comparing objective values.

  1. [1]:Lagrange multipliers
  2. [2]:Re-solving the disk problem with multipliers
  3. [3]:Closest and furthest points on a quartic curve
  4. [4]:Optimal Cobb-Douglas allocation

Minimise a distance through its square

For a fixed point p\mathbf{p}, minimise the squared distance

D(x)=∥x−p∥2D(\mathbf{x})=\|\mathbf{x}-\mathbf{p}\|^2

instead of the distance ∥x−p∥\|\mathbf{x}-\mathbf{p}\|. Since the square-root function is increasing on non-negative values, the points that minimise or maximise the two objectives are the same.

The squared expression avoids a square root and usually gives a polynomial objective. If the constraint has a visible normal direction, the shortest segment can sometimes be found geometrically before writing the multiplier system.

  1. [1]:Constrained extremum
  2. [2]:The closest point on a plane
  3. [3]:Closest and furthest points on a quartic curve
  4. [4]:The closest point on a parabola

On this page

Optimise subject to one equality constraintMinimise a distance through its square