Introduction to Optimization Problems · 优化问题简介
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| optimization/ˌɒptɪmaɪˈzeɪʃn/ | 最优化 | zuì yōu huà |
| variable/ˈveərɪəbl/ | 变量 | biàn liàng |
| objective function/ɒbˈdʒektɪv ˈfʌŋkʃn/ | 目标函数 | mù biāo hán shù |
| constraint/kənˈstreɪnt/ | 约束条件 | yuē shù tiáo jiàn |
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.
让某物尽可能大(或小)
- 现实问题会问:什么尺寸给出最大面积?最便宜成本?最短时间?
- 这些是最优化问题——求某个量的最大或最小。
- 微积分能解它们,因为极值就活在导数为零之处。
- 这一课讲建立模型;求解在下一课。
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.
命名变量并写出目标
- 为你能控制的量选一个变量(一段长度、一个半径、一个价格)。
- 把你想最大化或最小化的东西写成该变量的函数——目标函数。
- 例:对周长固定的矩形,把面积写成一条边长的函数。
- 画图并标注——大多数建模错误源于没标注的图。
In "maximize area with a fixed perimeter," the area is the... · 在“固定周长最大化面积”中，面积是...
You optimize the objective (area); the perimeter is the constraint. · 你优化的是目标（面积）；周长是约束条件。
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.
用约束化到一个变量
- 目标通常一开始有两个变量(如长和宽)。
- 一个约束方程把它们联系起来(固定周长、固定体积、预算)。
- 从约束解出一个变量并代入,使目标依赖于单个变量。
- 现在它就是一个普通的"求极值"问题。
Area as one downward parabola · 面积作为一条开口向下的抛物线
y = ax² + bx
After using the constraint, area becomes $A=10x-x^2$ — a downward parabola with a single peak to find. · 利用约束条件后，面积变为$A=10x-x^2$——一条开口向下的抛物线，只有一个峰值需要寻找。
A constraint equation is used to... · 约束方程用于...
Solve the constraint for one variable and substitute. · 解出约束中的一个变量并代入。
The goal of the setup is to write the objective in terms of a ____ variable. · 设置的目标是用一个____变量写出目标函数。
One variable ⇒ an ordinary extremum problem. · 一个变量 ⇒ 普通极值问题。
Order the optimization setup steps. · 排列优化设置的步骤。
Variable, objective, constraint, substitute. · 变量、目标、约束、代入。
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.
一个建模范例
- "一个农民有 $40$ 米栅栏,沿墙围一个矩形围栏(墙那侧不用栅栏)。"
- 约束:所用栅栏 $=2x+y=40$,所以 $y=40-2x$。
- 目标:面积 $A=xy=x(40-2x)=40x-2x^2$,现在是 $x$ 单独的函数。
- 准备最大化 $A(x)$——那是 5.11 课。
With $2x+y=40$ (wall on one side), the area $A=xy$ becomes... · 当$2x+y=40$（一侧有墙）时，面积$A=xy$变为...
$y=40-2x$, so $A=x(40-2x)=40x-2x^2$. · $y=40-2x$，所以 $A=x(40-2x)=40x-2x^2$。
Noting the domain of the variable (e.g. $0
The domain bounds where a valid answer can be. · 有效答案可能的边界。
Keep the objective (what you optimize) separate from the constraint (the fixed relationship). You maximize/minimize the objective, and you use the constraint only to eliminate a variable. Also note the domain of the variable (e.g. $x>0$ and $y>0$) — it bounds where a valid answer can lie.
把目标(你要优化的)与约束(固定的关系)分开。你最大化/最小化目标,而只用约束来消去一个变量。也要注意变量的定义域(如 $x>0$ 且 $y>0$)——它界定有效答案的范围。
Set up: maximize the area of a rectangle with perimeter $20$.
- Let the sides be $x$ and $y$. Constraint: $2x+2y=20\Rightarrow y=10-x$.
- Objective: $A=xy=x(10-x)=10x-x^2$.
- Domain: $0
建模:最大化周长为 $20$ 的矩形面积。
- 设两边为 $x$ 和 $y$。约束:$2x+2y=20\Rightarrow y=10-x$。
- 目标:$A=xy=x(10-x)=10x-x^2$。
- 定义域:$0
An optimization problem finds a maximum or minimum. Set it up by naming a variable, writing the objective function to optimize, and using a constraint equation to reduce it to a single variable. Note the domain. Then it becomes a standard extremum problem (next lesson).
最优化问题求最大或最小。建模方式:命名一个变量,写出要优化的目标函数,用一个约束方程把它化到单个变量。注意定义域。然后它就变成一个标准的极值问题(下一课)。