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It works by solving the non-integer linear program, then testing if the Optimum found is also an integer solution. If this is not the case, a new restriction is added that cuts off the non-integer solution but does not cut off any integer points of the Feasible Region . This is repeated until an optimal integer solution is found.

Interpreted geometrically, a restriction is equivalent to an oriented Hyperplane , allowing only solutions on one side of the plane.


GOMORY'S CUT


We have an admissible solution x and we have a base B associated to x that

:\begin{bmatrix}B & F \end{bmatrix}\begin{bmatrix}x_b \ x_f \end{bmatrix}=b \Rightarrow Bx_b+Fx_f =b\Rightarrow

:\Rightarrow x_b=B^{-1}b-B^{-1}Fx_f=\bar b\ - \bar A\ x_f \Rightarrow x_b+ \bar A\ x_f= \bar b\

If we have a fractional solution so we have a nth element of fractionated x .

:(1)\,

::x_n + \sum_{j \in\ \zeta\ }^N \bar a_{n,j}x_j= \bar b_n

::\begin{bmatrix} \ x_b \ \ \end{bmatrix} + \begin{bmatrix} & &\ & \bar A\ & \ & &\end{bmatrix} \begin{bmatrix} x_f \end{bmatrix} = \begin{bmatrix} \ \bar b\ \ \ \end{bmatrix}

:(2) \,

:: x_n + \sum_{j \in\ \zeta\ }^N \left \lfloor \bar a_{n,j}x_j ight floor \le \; \left \lfloor \bar b_n ight floor

:: is a cut or '''integer formulation of Gomory's cut'''

Cutting plane methods are also applicable in Nonlinear Programming . The underlying principle is to approximate the Feasible Region of a nonlinear (convex) program by a finite set of closed half spaces and to solve a sequence of approximating Linear Program s.


REFERENCES


Avriel, Mordecai (2003). Nonlinear Programming: Analysis and Methods. Dover Publishing. ISBN 0-486-43227-0


SEE ALSO