Gomory’s cutting plane algorithm for integer programming Prepared by Shin-ichi Tanigawa

Rounding does not give any useful result

We first solve the LP-relaxation

Optimize using primal simplex method

The optimal solution is fractional

(1) Generating an objective row cut

(weakening) (1) Generating an objective row cut

(weakening) (1) (2) (for integers) Generating an objective row cut

(weakening) (1) (2) (for integers) (2) - (1) Generating an objective row cut Cutting plane is violated by current optimum solution

(weakening) (1) (2) (for integers) (2) - (1) (substitute for slacks) Generating an objective row cut

The first cutting plane:

A new slack variable is added:

The new cut is added to the dictionary

Re-optimize using dual simplex method

A new fractional solution has been found

(1) Generating a constraint row cut

(1) (weaken) Generating a constraint row cut

(1) (2) Generating a constraint row cut (valid for integers)

(1) (2) (2) – (1) Generating a constraint row cut

(1) (2) (2) – (1) Generating a constraint row cut

The second cutting plane

Add a new slack variable:

The new cut is inserted into the optimum dictionary

Re-optimize using dual simplex method

The new optimum solution is integral