Lagrange Multipliers

TL;DR

• A method for finding maxima or minima of a function when one or more constraints must be satisfied.
• Combine the objective function and constraint(s) using a Lagrange multiplier (λ), then solve the resulting equations for the optimal variables.

Definition

Lagrange multipliers are a mathematical tool used to find the maximum or minimum value of a function subject to constraints.

Explanation

Lagrange multipliers work by incorporating constraint equations into the optimization of an objective function. The typical procedure is:

• Define the objective function to be maximized or minimized.
• Write the constraint equation(s) that the variables must satisfy.
• Combine the objective and constraint(s) into a single equation using a Lagrange multiplier (λ).
• Solve the resulting equations to obtain the optimal values of the variables.

The method therefore finds an optimal solution by combining multiple equations and constraints into one system to be solved.

Examples

Rectangular field with fixed perimeter

Objective function (area): $A = lw$ Constraint (fixed fencing of 100 feet): $2l + 2w = 100$ Combine objective and constraint with a Lagrange multiplier: $A - \lambda(2l + 2w - 100) = 0$ Solving this equation gives the optimal dimensions: $l = 25,\quad w = 20$ Area: $A = 500\ \text{square feet}$

Shortest distance between two fixed points

Objective function (distance between two points): $d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$ Constraint (points fixed at ((2,3)) and ((5,7))): $x_1 = 2,\ y_1 = 3,\ x_2 = 5,\ y_2 = 7$ Combine objective and constraint with Lagrange multipliers: $d - \lambda(x_1 - 2) - \lambda(y_1 - 3) - \lambda(x_2 - 5) - \lambda(y_2 - 7) = 0$ Solving yields the shortest distance: $d = \sqrt{3^2 + 4^2} = 5$

Related terms

• Constraint
• Objective function
• Lagrange multiplier (λ)