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Optimization

Find the best answer when every choice changes the trade-off.

Follow the mechanism1 / 3

A goal competes with a constraint. Changing one choice helps one part of the design while forcing another part to give something up.

fixed perimeterP = 20 · area 21.0
Focus the idea
A rectangle reaches maximum area when its sides are equalA rectangle has fixed perimeter 20. Its width is 3, length is 7.0, and area is 21.0. The area curve has a maximum at width 5.0, where the derivative is zero.area A(w)width wA = 21.0maximum at w = 5.0width 3length 7.0A(w) = w(10 − w)
width w3.0
length7.0
area A(w)21.0
slope A′(w)4.0

The slope is positive, so increasing width still increases area. Move toward w = 5.0.

Optimization turns a goal and a constraint into one function. The derivative identifies where improvement stops; endpoints and other candidates still need comparison before calling it a global best.