[mathematical modeling] summary of common basic models

Linear programming （Linear Programming)

• An important branch of operations research —— Mathematical programming . Linear programming is limited by a set of linear constraints , The problem of finding the maximum or minimum of a linear objective function .
• Concept ： Feasible solution 、 Optimal solution 、 The feasible region .
• Matlab The order for solving linear programming in is as follows ,x Returns the value of the decision vector ;fval Returns the optimal value of the objective function ;f It's the value vector .

[x,fval] = linprog(f,A,b,Aeq,beq,lb,ub)

• Many problems that don't seem to be linear programming , It can also be solved by transforming the problem into linear programming .
  For example, for the following mathematical programming problems ：
  $$\begin{array}{c}\min \left|x_1\right|+\left|x_2\right|+\cdots +\left|x_n\right|\\ \text{ s.t. }\mathbf{A}\mathbf{x}⩽{\mathbf{b}}_{\circ }\end{array}$$
  Just notice the facts ：
  For any $x_i$, There is $u_i,v_i\ge 0$ Satisfy $x_i=u_i-v_i,\left|x_i\right|=u_i+v_i$
  So take $u_i=\frac{x_i+|x_i|}{2},v_i=\frac{|x_i|-x_i}{2}$, remember $\mathbf{u}={\left[u_1,\cdots ,u_n\right]}^{\mathrm{T}},\mathbf{v}={\left[v_1,\cdots ,v_n\right]}^{\mathrm{T}}$,
  So as to rewrite the model into ：
  $$\begin{array}{ll}\min & {\sum }_{i=1}^n\left(u_i+v_i\right),\\ \text{ s.t. }& \{\begin{array}{l}[\mathbf{A},-\mathbf{A}]\left[\begin{array}{l}\mathbf{u}\\ \mathbf{v}\end{array}\right]⩽\mathbf{b},\\ \mathbf{u},\mathbf{v}⩾0.\end{array}\end{array}$$