If X1 and X2 are the optimal solutions of a linear programming problem, it is proved that all the points on the line between the two points are also the optimal solutions of the problem

If X1 and X2 are the optimal solutions of a linear programming problem, it is proved that all the points on the line between the two points are also the optimal solutions of the problem

Go to the operations research textbook,
Tsinghua University Third Edition "operations research", from 16 also see
First look at the basic concepts: convex set, convex combination, vertex
Let's look at the following lemmas
Very simple, linear programming has a solution, the solution set must be convex, x1, X2 are two vertices, any point on the line of two points can be expressed as a convex combination of two points, since X1 and X2 are the optimal solution, then their convex combination must be the optimal solution
Do not understand the principle of reading, the idea is probably like this

The simplex method is used to solve the following linear programming problems
max z ＝100x1＋200x2
st.x1＋ x2≤500
x1 ≤200
2x1＋6x2≤1200
x1,x2≥0

The optimal solution is: X1 = 200; x2 = 133.333
Objective function value of optimal solution: z = 33333.3
It has been verified by programming

Using graphic method and simplex to solve linear programming problems. Max z = 2x1 + x2 St {3x1 + 5x2}

Only 2 unknowns, graphic method to draw their own
Simplex:
Standard type: maxz = 2x1 + x2 + 0x3 + 0x4
ST:3X1+5X2+X3=15
6X1+2X2+X4=24
Cj→ 2 1 0 0
CB based B X1 x2 X3 X4
0 X3 15 3 5 1 0
0 X4 24 [6] 2 0 1
The number of tests is 2 100
-------------------------------------------------------
0 X3 3 0 [4] 1 -1/2
2 X1 4 1 1/3 0 1/6
Inspection number 0 1 / 3 0 - 1 / 3
--------------------------------------------------------
1 X2 3/4 0 1 1/4 -1/8
2 X1 2/9 1 0 -1/12 145/24
Test number 0 0 - 1 / 12 - 17 / 36
--------------------------------------------------------
So x = (2 / 9 3 / 4 00)
maxz=43/36

Use graphic method to solve the following linear programming (20 points) max z = 6X1 + 4x2 s.t.2x1 + 3x2 ≤ 100 4x1 + 2x2 ≤ 120 x1, X2 ≥ 0

LZ topic is not easy