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【Weekly Diary: Mathematical Modeling Learning (7)】
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Basic Principles and Programming Practices of Nonlinear Programming
Nonlinear Programming Fundamentals and Programming Practice
One. Conception.
I. Concept
Definition: A planning problem in which the objective function or constraint contains a nonlinear function
Definition: A planning problem where the objective function or constraint conditions contain nonlinear functions.
Two. instance
II. Example
Investment Decision Questions:
Investment Decision Problem：
Best investment purpose: the minimum investment amount and the maximum total return.
Objective: To minimize the investment amount while maximizing the total return.
bi is the ith income; AI is the ith investment amount; A is the total funding.
bi represents the return of the i-th investment; ai represents the investment amount of the i-th investment; A represents the total capital.
Objective function: Maximize the ratio of total return to total investment.
Objective function: Maximize the ratio of total return to total investment.
Constraints:
Constraint conditions:
- The sum of the amount of each investment is less than the total funds.
A. The sum of investment amounts for each investment is less than the total capital.
- Restriction is used to restrict Xi to only 0 and 1.
B. A restrictive condition limits Xi to only take values of 0 and 1.
Cons: No consideration of risk
Drawback: Risk is not considered.
Three. Mathematical models of nonlinear programming
III. Mathematical Model of Nonlinear Programming
A*x≤b and Aeq*x=beq are constraints for linear programming.
Ax ≤ b and Aeqx = beq are the constraint conditions of linear programming.
c(x)≤0 is a constraint for nonlinear programming.
c(x) ≤ 0 is the constraint condition for nonlinear programming.
ceq(x)=0 is a constraint for integer programming.
ceq(x) = 0 is the constraint condition for integer programming.
lb≤x≤ub is a restriction on x.
lb ≤ x ≤ ub are the limitations on x.
Among them, it is important to note that:
Note:
fun needs to be defined as a nonlinear function in MATLAB.
fun needs to be defined as a nonlinear function in MATLAB.
nonlcon is a defined nonlinear vector function, the same as c(x)≤0, which needs to be defined with the help of function.
nonlcon is the defined nonlinear vector function, similar to c(x) ≤ 0, which requires function definition using the function keyword.
options一般不用写。
options are generally not necessary to write.
Four. Programming examples
IV. Programming Example
Topic:
Problem Statement:
Steps:
Solution Steps:
fun1（x）是目标函数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与x1+x2*x2+x3*x3≤20均需要换成c（x）≤0的形式，为非线性约束条件。
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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Reference: Bilibili
Translation: Wenxin Yiyan