Mixed-integer cuts: Difference between revisions
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== Cutting Planes == | == 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. | 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 [1]. | |||
<math> | |||
a_{i,j} x_j \leq b_i</math> | |||
For a given knapsack inequality: | |||
jaijxjbi xj{0,1} | |||
The gomory cut is defined as: | |||
j⌊aij⌋xj⌊bi⌋ | |||
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 nonintegerb∗iconstraint:∑ja∗ijxj=b∗i | |||
3. Rewrite constraint using fractional partsfij=aij−[aij],fi=bi−[bi]:∑jf∗ijxj−f∗j= [b∗j]−∑j[a∗ij]xj | |||
Heuristic for step 2: chooseb∗iwith largestf∗i. | |||
4. Add new constraint∑jfijxj−fj≥0, with integer excess, to tableau. | |||
5. Repeat steps 2-4 (using dual simplex) until all rhsb∗i’s are integers. | |||
3 x1 + 3 2/5 x2 - 2/5 x3= 8 ¾ | |||
Cut: | |||
2/5 x2 + 3/5 x3 >= ¾ | |||
-3 1/4 x1 + 2/5 x2 - 2/5 x3= 7 ⅚ | |||
Cut: | |||
3/4 x1 + 2/5 x2 + 3/5 x3 >= 5/6 | |||
== Cover Cuts == | |||
For a given knapsack inequality: | |||
jaijxjbi xj{0,1} | |||
Let CJ and jCaj>b | |||
The cover inequality is: | |||
jCxjC-1, xj{0,1} | |||
Example: | |||
Change numbers | |||
Maximize Z = 11x1+6x2+6x3+5x4+5x5+4x6+x7<=19 | |||
Some minimal cover inequalities of Z are: | |||
X1+x2+x3 <=2 | |||
X1+x2+x6 <=2 | |||
X1+x5+x6 <= 2 | |||
X3+x4+x5+x6 <=3 | |||
== Applications == | |||
== 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] Gomory Cuts revisited | |||
Mixed integer nonlinear programming | |||
Laurence A. Wolsey - Integer Programming |
Revision as of 17:52, 21 November 2020
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 [1].
For a given knapsack inequality:
jaijxjbi xj{0,1}
The gomory cut is defined as:
j⌊aij⌋xj⌊bi⌋
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 nonintegerb∗iconstraint:∑ja∗ijxj=b∗i
3. Rewrite constraint using fractional partsfij=aij−[aij],fi=bi−[bi]:∑jf∗ijxj−f∗j= [b∗j]−∑j[a∗ij]xj
Heuristic for step 2: chooseb∗iwith largestf∗i.
4. Add new constraint∑jfijxj−fj≥0, with integer excess, to tableau.
5. Repeat steps 2-4 (using dual simplex) until all rhsb∗i’s are integers.
3 x1 + 3 2/5 x2 - 2/5 x3= 8 ¾
Cut:
2/5 x2 + 3/5 x3 >= ¾
-3 1/4 x1 + 2/5 x2 - 2/5 x3= 7 ⅚
Cut:
3/4 x1 + 2/5 x2 + 3/5 x3 >= 5/6
Cover Cuts
For a given knapsack inequality:
jaijxjbi xj{0,1}
Let CJ and jCaj>b
The cover inequality is:
jCxjC-1, xj{0,1}
Example:
Change numbers
Maximize Z = 11x1+6x2+6x3+5x4+5x5+4x6+x7<=19
Some minimal cover inequalities of Z are:
X1+x2+x3 <=2
X1+x2+x6 <=2
X1+x5+x6 <= 2
X3+x4+x5+x6 <=3
Applications
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] Gomory Cuts revisited
Mixed integer nonlinear programming
Laurence A. Wolsey - Integer Programming