In the context of the Traveling salesman problem on three nodes, this rather weak[ why? In mathematical optimization , the cutting-plane method is any of a variety of optimization methods that iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are commonly used to find integer solutions to mixed integer linear programming MILP problems, as well as to solve general, not necessarily differentiable convex optimization problems. Cutting plane methods for MILP work by solving a non-integer linear program, the linear relaxation of the given integer program. The theory of Linear Programming dictates that under mild assumptions if the linear program has an optimal solution, and if the feasible region does not contain a line , one can always find an extreme point or a corner point that is optimal. The obtained optimum is tested for being an integer solution.
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Gomory in the s as a method for solving integer programming and mixed-integer programming problems. However most experts, including Gomory himself, considered them to be impractical due to numerical instability, as well as ineffective because many rounds of cuts were needed to make progress towards the solution. Things turned around when in the mids Cornuejols and co-workers showed them to be very effective in combination with branch-and-cut and ways to overcome numerical instabilities.
Gomory cuts, however, are very efficiently generated from a simplex tableau, whereas many other types of cuts are either expensive or even NP-hard to separate. Let an integer programming problem be formulated as The method proceeds by first dropping the requirement that the xi be integers and solving the associated linear programming problem to obtain a basic feasible solution.
Geometrically, this solution will be a vertex of the convex polytope consisting of all feasible points. If this vertex is not an integer point then the method finds a hyperplane with the vertex on one side and all feasible integer points on the other. This is then added as an additional linear constraint to exclude the vertex found, creating a modified linear programming program.
The new program is then solved and the process is repeated until an integer solution is found. Rewrite this equation so that the integer parts are on the left side and the fractional parts are on the right side: For any integer point in the feasible region the right side of the this equation is less than 1 and the left side is an integer, therefore the common value must be less than or equal to 0.
So the inequality must hold for any integer point in the feasible region. Furthermore, if ci is not an integer for the basic solution x, So the inequality above excludes the basic feasible solution and thus is a cut with the desired properties.