The cutting plane method is mainly used to solve integer programming problems. 1958 was put forward by Gomorrah, USA. The basic idea is to solve the corresponding linear programming problem without considering integer constraints. If the optimal solution of linear programming problem happens to be an integer solution, then this solution is the optimal solution of integer programming problem. Otherwise, a new constraint is added, called the cutting plane.
The cutting plane must have two properties: at least it is a non-integer optimal solution cut from the feasible region of linear programming problems; Without cutting off any feasible region of integers, we continue to solve the linear programming problem on the reduced feasible region. By repeating the above practice, the optimal solution of the integer programming problem can be obtained at the integer poles of the reduced feasible region after finite cuts.
The cutting plane method is a relatively simple method to solve integer programming, which was put forward by American scholar R.E.GoMory in 1958. Its basic idea is basically the same as the branch-and-bound method, that is, the simplex method is used to solve the corresponding linear programming without considering the integer constraint of variables.
If the optimal solution is an integer solution, it is also the optimal solution of the original integer programming problem. 3 If the optimal solution is not an integer solution, the branch-and-bound method divides the original integer programming into two branches by arbitrarily selecting a variable Xk=bk with a fractional value.
Its essence is to divide the original feasible region R into two parallel planes Xk=[bk] and Xk=[bk]+ 1 and R2 perpendicular to the coordinate axis, and to reduce the feasible region by removing the part of the feasible region between the two parallel planes that does not contain integer solutions.
The cutting plane method was put forward by RalphGomory in 1950s, which was used to solve integer programming and mixed integer programming. But most experts at that time, including Gomory himself, thought that this method was of no practical value because of its unstable numerical value. At the same time, this method may not be effective because of the need for multiple rounds of cutting in the solution process.