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【周周記：數學模組化學習（7）】
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Dear, this is LearningYard New Academy.
Today, the editor brings you
"Weekly Diary: Learning Mathematical Modeling (7)"
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非線性規劃基本原理與程式設計實踐
Nonlinear Programming Fundamentals and Programming Practice
一.概念.
I. Concept
定義：目标函數或限制條件中包含非線性函數的規劃問題
Definition: A planning problem where the objective function or constraint conditions contain nonlinear functions.
二.執行個體
II. Example
投資決策問題：
Investment Decision Problem：
最佳投資目的：投資額最小而總收益最大。
Objective: To minimize the investment amount while maximizing the total return.
bi為第i筆的收益；ai為第i筆的投資額；A為總資金。
bi represents the return of the i-th investment; ai represents the investment amount of the i-th investment; A represents the total capital.
目标函數：極大化總收益和總投資之比。
Objective function: Maximize the ratio of total return to total investment.
限制條件：
Constraint conditions:
- 每筆投資金額之和小于總資金。
A. The sum of investment amounts for each investment is less than the total capital.
- 限制條件用于限制Xi隻能取0與1 。
B. A restrictive condition limits Xi to only take values of 0 and 1.
缺點：未考慮風險
Drawback: Risk is not considered.
三.非線性規劃的數學模型
III. Mathematical Model of Nonlinear Programming
A*x≤b和Aeq*x=beq是線性規劃的限制條件。
Ax ≤ b and Aeqx = beq are the constraint conditions of linear programming.
c（x）≤0是非線性規劃的限制條件。
c(x) ≤ 0 is the constraint condition for nonlinear programming.
ceq（x）=0是整數規劃的限制條件。
ceq(x) = 0 is the constraint condition for integer programming.
lb≤x≤ub是對x的限制條件。
lb ≤ x ≤ ub are the limitations on x.
其中需要注意的是：
Note:
fun在MATLAB中需要進行非線性函數的定義。
fun needs to be defined as a nonlinear function in MATLAB.
nonlcon是定義的非線性向量函數，同c（x）≤0，需要進行函數定義，借助function。
nonlcon is the defined nonlinear vector function, similar to c(x) ≤ 0, which requires function definition using the function keyword.
options一般不用寫。
options are generally not necessary to write.
四.程式設計執行個體
IV. Programming Example
題目：
Problem Statement:
解題步驟：
Solution Steps:
fun1（x）是目标函數f（x）
fun1(x) represents the objective function f(x)
sum（x.^2）=x1*x1+x2*x2+x3*x3
x1*x1-x2+x3*x3≥0與x1+x2*x2+x3*x3≤20均需要換成c（x）≤0的形式，為非線性限制條件。
sum(x.^2) = x1x1 + x2x2 + x3x3
x1x1 - x2 + x3x3 ≥ 0 and x1 + x2x2 + x3*x3 ≤ 20 need to be converted into the form of c(x) ≤ 0, which are the nonlinear constraint conditions.
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