Summary of Lagrange Multipliers

Essential Concepts

  • An objective function combined with one or more constraints is an example of an optimization problem.
  • To solve optimization problems, we apply the method of Lagrange multipliers using a four-step problem-solving strategy.

Key Equations

  • Method of Lagrange multipliers, one constraint
    [latex]\begin{array}{cc} \nabla f(x_{0},y_{0}) &= \lambda\nabla g(x_{0},y_{0}) \\ \hfill g(x_{0},y_{0}) &= 0 \hfill \end{array}[/latex]
  • Method of Lagrange multipliers, two constraints
    [latex]\begin{array}{cc} \nabla f(x_{0},y_{0},z_{0}) &= \lambda_{1}\nabla g(x_{0},y_{0},z_{0})+\lambda_{2}\nabla h(x_{0},y_{0},z_{0}) \\ \hfill g(x_{0},y_{0},z_{0}) &= 0 \hfill\\ \hfill h(x_{0},y_{0},z_{0}) &= 0 \hfill \end{array}[/latex]

Glossary

constraint
an inequality or equation involving one or more variables that is used in an optimization problem; the constraint enforces a limit on the possible solutions for the problem
Lagrange Multiplier
the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variable [latex]\lambda[/latex]
method of Lagrange multipliers
a method of solving an optimization problem subject to one or more constraints
objective function
the function that is to be maximized or minimized in an optimization problem
optimization problem
calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers