Solve the problem of minimizing the surface area of a cylinder of given value V. The two design variables are the radius and height. The equality constraint is the volume constraint. (solution)

2

2

2 2Minimize r rhSubject to

V r h

π π

π

+

=

Using a Lagrange multiplier, the objective function is reformulated as ( )2 2

2

2 2

1/3

1/3 1/3 1/3

2 2

4 2 2 0 2 0

2 0 2

0

4 2stationary point:2

4 40.5 44

L r rh r h V

L Lr h rh r h rhr rL Lr r rh hL L Vr h V r h

V hh rr

V Vh rV

π π λ π

π π π λ λ

π π λ λ

πλ λ π

λπ

πλπ π

= + + −

∂ ∂= + + = ⇒ + + =

∂ ∂∂ ∂

= + = ⇒ = −∂ ∂∂ ∂

= − = ⇒ =∂ ∂

= = = −

= = = −

There is one stationary point. Also both design variables must be greater than zero to be physical.