Sort: unconstrained optimization or constrained (Lagrange multipliers)?

Groups: Unconstrained · Constrained (Lagrange)

Minimize $f(x,y)$ with no restrictions
Solve $\nabla f$ = $\lambda \nabla g$ and $g = 0$
Apply the second derivative test
Maximize $f(x,y)$ subject to $g(x,y) = 0$
Find where $\nabla f$ = 0
Optimize f on the boundary of a region

Hints

Ask whether the variables are free or tied by an equation.
An equality relation between the variables signals the constrained method.

Show the answer

Unconstrained: Minimize $f(x,y)$ with no restrictions, Find where $\nabla f$ = 0, Apply the second derivative test

Constrained (Lagrange): Maximize $f(x,y)$ subject to $g(x,y) = 0$ , Optimize f on the boundary of a region, Solve $\nabla f$ = $\lambda \nabla g$ and $g = 0$

Why

Lagrange multipliers handle optimization with equality constraints.

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Sort: unconstrained optimization or constrained (Lagrange multipliers)?

Practise Optimization: Critical Points and Lagrange Multipliers

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