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