Mixed-integer cuts

Author: Ryan Carr, Patrick Guerrette, Mark James (SysEn 5800 Fall 2020)

Introduction

In mixed-integer programming, mixed-integer cuts are additional constraints placed upon the problem in order to make the extreme points of the feasible region be integers as opposed to points with fractional values. These cuts reduce the feasible region, making the problem easier to solve. A mixed-integer problem can be reduced with mixed-integer cuts until its feasible region reaches the convex hull, where all extreme points of the feasible region are integers.

Cutting Planes

The process to create cuts is to first take the mixed-integer problem and then relax the variables to a linear programming problem and tighten the problem through additional constraints such that extreme points of the feasible region are integers.

Gomory Cuts

Ralph Gomory sought out to solve mixed integer linear programming problems by using cutting planes in the late fifties and early sixties .

For a given knapsack inequality:

$\sum _{j}a_{i,j}x_{j}\leq b_{i}\qquad x_{j}\in \{0,1\}$

The Gomory cut is defined as:

$\sum _{j}\lfloor a_{i,j}\rfloor x_{j}\leq \lfloor b_{i}\rfloor$

Using the simplex method with Gomory cuts(fractional example):

1. Begin with LP in standard form for application of simplex method.

2. Apply simplex method until convergence, and select any non-integer $b_{i}^{*}$constraint:

$\sum _{j}a_{i,j}^{*}x_{j}=b_{i}^{*}$

3. Rewrite constraint using fractional parts $f_{i,j}=a_{i,j}-[a_{i,j}],\quad f_{i}=b_{i}-[b_{i}]$:

$\sum _{j}f_{i,j}^{*}x_{j}-f_{j}^{*}=b_{j}^{*}=[b_{j}^{*}]-\sum _{j}[a_{i,j}^{*}]x_{j}$

4. Add new constraint$\sum _{j}f_{i,j}x_{j}-f_{j}\geq 0$, with integer excess, to tableau.

5. Repeat steps 2-4 until all right hand side $b_{i}^{*}$'s are integers.

Example:

$3x_{1}+4{\frac {1}{5}}x_{2}-{\frac {3}{5}}x_{3}=8{\frac {1}{4}}$

Cut:

${\frac {4}{5}}x_{2}+{\frac {2}{5}}x_{3}\geq {\frac {1}{4}}$

$-3{\frac {3}{4}}x_{1}+2{\frac {3}{5}}x_{2}-1{\frac {1}{5}}x_{3}=7{\frac {5}{6}}$

Cut:

${\frac {1}{4}}x_{1}+{\frac {3}{5}}x_{2}+{\frac {4}{5}}x_{3}\geq {\frac {5}{6}}$

Cover Cuts

The feasible region of a knapsack problem can be reduced using minimal cover inequalities. The short coming of the cut is that it does not reflect the weights of each item in the knapsack problem because the coefficients of the inequalities derived from the knapsack problem are fixed to 1 {ref}2{/ref}.

For a given knapsack inequality:

$\sum _{j}a_{i,j}x_{j}\leq b_{i}\qquad x_{j}\in \{0,1\}$

Let $C\subset J$ and $\sum _{j\in C}a_{j}>b$

The cover inequality is:

$\sum _{j\in C}x_{j}\leq |C|-1,\quad x_{j}\in \{0,1\}$

Example:

Change numbers

$\max Z=12x_{1}+5x_{2}+7x_{3}+8x_{4}+3x_{5}+5x_{6}+6x_{7}\leq 24$

Some minimal cover inequalities of Z are:

$x_{1}+x_{2}+x_{4}\leq 2$

$x_{1}+x_{3}+x_{4}\leq 2$

$x_{1}+x_{4}+x_{6}\leq 2$

$x_{2}+x_{3}+x_{4}+x_{6}\leq 3$

Conclusion

Mixed Integer Cuts allows for shorter computational time in solving mixed integer linear programs by refining the feasible region with linear inequalities. If the optimum found by solving the non-integer linear program is non-integer, a linear inequality can be determined to remove the solution from the feasible region leading to the convex hull.

References

1. Balas, E., et al. “Gomory Cuts Revisited.” Operations Research Letters, vol. 19, no. 1, July 1996, pp. 1–9., doi:10.1016/0167-6377(96)00007-7.

2. Weismantel, Robert. “On the 0/1 Knapsack Polytope.” Mathematical Programming, vol. 77, no. 3, 1997, pp. 49–68., doi:10.1007/bf02614517.