Key Equations

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

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 $\lambda$

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