Giới thiệu về Các bài toán Tối ưu hóa

Making something as big (or small) as possible

• Real problems ask: what dimensions give the largest area? the cheapest cost? the shortest time?
• These are optimization 最优化 problems — find the max or min of a quantity.
• Calculus solves them because extrema live where the derivative is zero.
• This lesson is about the setup; the solving comes next.

Name a variable and write the objective

• Choose a variable 变量 for the quantity you can control (a length, a radius, a price).
• Write the thing you want to maximize or minimize as a function of that variable — the objective function 目标函数.
• Example: for a rectangle of fixed perimeter, write the area in terms of one side length.
• Draw a picture and label it — most setup errors are unlabeled diagrams.

Fenced enclosures: optimization finds the dimensions that maximise area for a fixed fence length

Use a constraint to get one variable

• Usually the objective starts with two variables (like length and width).
• A constraint 约束条件 equation relates them (a fixed perimeter, a fixed volume, a budget).
• Solve the constraint for one variable and substitute, so the objective depends on a single variable.
• Now it's an ordinary "find the extremum" problem.

A worked setup

• "A farmer has $40$ m of fence for a rectangular pen against a wall (no fence on the wall side)."
• Constraint: fence used $=2x+y=40$, so $y=40-2x$.
• Objective: area $A=xy=x(40-2x)=40x-2x^2$, now a function of $x$ alone.
• Ready to maximize $A(x)$ — that's lesson 5.11.