Mixed-integer cuts: Difference between revisions
(Added Gomory Cuts, Cover Cuts, Conclusion, and References) |
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Ralph Gomory sought out to solve mixed integer linear programming problems by using cutting planes in the late fifties and early sixties [1]. | 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: | For a given knapsack inequality: | ||
<math> | |||
\sum_{j}a_{i,j} x_j \leq b_i \qquad x_j \in \{0,1\} </math> | |||
The gomory cut is defined as: | The gomory cut is defined as: | ||
<math> | |||
\sum_{j} \lfloor a_{i,j} \rfloor x_j \leq \lfloor b_i \rfloor </math> | |||
Using the simplex method with gomory cuts(fractional example): | Using the simplex method with gomory cuts(fractional example): | ||
Line 24: | Line 24: | ||
1. Begin with LP in standard form for application of simplex method. | 1. Begin with LP in standard form for application of simplex method. | ||
2. Apply simplex method until convergence, and select any | 2. Apply simplex method until convergence, and select any noninteger <math> | ||
b_i^* </math>constraint: | |||
<math> | |||
\sum_{j}a_{i,j}^*x_j=b_i^* </math> | |||
3. Rewrite constraint using fractional parts <math> | |||
f_{i,j}=a_{i,j}-[a_{i,j}], \quad f_i=b_i - [b_i] </math>: | |||
<math> | |||
\sum_{j}f_{i,j}^*x_j-f_j^*=b_j^*=[b_j^*]-\sum_{j}[a_{i,j}^*]x_j </math> | |||
4. Add new constraint<math> | |||
\sum_{j}f_{i,j}x_j-f_j\geq0 </math>, with integer excess, to tableau. | |||
5. Repeat steps 2-4 until all right hand side<math> | |||
b_i^*'s </math> are integers. | |||
Example: | |||
3 x1 + 3 2/5 x2 - 2/5 x3= 8 ¾ | 3 x1 + 3 2/5 x2 - 2/5 x3= 8 ¾ |
Revision as of 18:31, 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:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{j}a_{i,j} x_j \leq b_i \qquad x_j \in \{0,1\} }
The gomory cut is defined as:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 noninteger Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_i^* } constraint:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{j}a_{i,j}^*x_j=b_i^* }
3. Rewrite constraint using fractional parts Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{i,j}=a_{i,j}-[a_{i,j}], \quad f_i=b_i - [b_i] } :
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{j}f_{i,j}^*x_j-f_j^*=b_j^*=[b_j^*]-\sum_{j}[a_{i,j}^*]x_j }
4. Add new constraintFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{j}f_{i,j}x_j-f_j\geq0 } , with integer excess, to tableau.
5. Repeat steps 2-4 until all right hand sideFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_i^*'s } are integers.
Example:
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