Mixed-integer cuts - Revision history

Pwg48: /* Numerical Example */

2020-12-14T03:07:14Z

Numerical Example

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 <math>s_1 = 5, s_2 = 2
 </math>
 {| class="wikitable"
 |+Simplex Tableau Iteration 1

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References

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	7. Gu, Zonghao, et al. “Lifted Flow Cover Inequalities for Mixed 0-1 Integer Programs.” Mathematical Programming, vol. 85, no. 3, 1999, pp. 439–467., doi:10.1007/s101070050067.		7. Gu, Zonghao, et al. “Lifted Flow Cover Inequalities for Mixed 0-1 Integer Programs.” Mathematical Programming, vol. 85, no. 3, 1999, pp. 439–467., doi:10.1007/s101070050067.

			8. Günlük, Oktay, and Yves Pochet. “Mixing Mixed-Integer Inequalities.” Mathematical Programming, vol. 90, no. 3, 2001, pp. 429–457., doi:10.1007/pl00011430.

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Gomory Cuts

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	== Gomory Cuts ==		== 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).		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:
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	<math>\frac{1}{4} x_1 + \frac{3}{5} x_2 + \frac{4}{5} x_3 \geq \frac{5}{6}</math>		<math>\frac{1}{4} x_1 + \frac{3}{5} x_2 + \frac{4}{5} x_3 \geq \frac{5}{6}</math>

	== Cover Cuts==		== 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 (2).		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 (2).

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 Where <math> l = \sum_{j \in C+} m_j - d > 0 </math> and
 <math> C++ = \{j \in C+ : m_j > l\} </math>
 == Numerical Example==

2020-12-14T02:19:01Z

Flow Cover Cut

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	<math>x_2+x_3+x_4+x_6 \leq 3</math>		<math>x_2+x_3+x_4+x_6 \leq 3</math>
	== Flow Cover Cut==		== Flow Cover Cut==
	Flow Cover Cuts were introduced and ~~derivced~~ by Padberg, Van Roy, and Wolsey(7).		Flow Cover Cuts were introduced and derived by Padberg, Van Roy, and Wolsey (7).

	Given the system		Given the system
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	where <math> N = N+ \cup N- </math> and <math> n = \|N\| </math>		where <math> N = N+ \cup N- </math> and <math> n = \|N\| </math>

			The <math> x </math> variables are flows that are constrained by the conservation inequality with demand for <math> d</math> and upper bound constraints, where <math> m_j </math> is the capacity of <math>j</math> and <math> y_j </math> is a binary variable that determing if <math> j </math> is open (7).

	The flow cover inequality is defined as:		The flow cover inequality is defined as:

2020-12-14T02:08:51Z

Flow Cover Cut

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	<math>x_2+x_3+x_4+x_6 \leq 3</math>		<math>x_2+x_3+x_4+x_6 \leq 3</math>
	== Flow Cover Cut==		== Flow Cover Cut==
	Flow Cover Cuts were introduced and derivced by Padberg, Van Roy, and Wolsey(7)		Flow Cover Cuts were introduced and derivced by Padberg, Van Roy, and Wolsey(7).

	Given the system		Given the system

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References

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	6. Lindsay Siegmundt, Peter Haddad, Chris Babbington, Jon Boisvert, Haris Shaikh. (2020). Branch and cut. December 2020. doi:[https://optimization.cbe.cornell.edu]/index.php?title=Branch_and_cut		6. Lindsay Siegmundt, Peter Haddad, Chris Babbington, Jon Boisvert, Haris Shaikh. (2020). Branch and cut. December 2020. doi:[https://optimization.cbe.cornell.edu]/index.php?title=Branch_and_cut

			7. Gu, Zonghao, et al. “Lifted Flow Cover Inequalities for Mixed 0-1 Integer Programs.” Mathematical Programming, vol. 85, no. 3, 1999, pp. 439–467., doi:10.1007/s101070050067.

2020-12-14T02:02:46Z

Flow Cover Cut

← Older revision		Revision as of 22:02, 13 December 2020
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	<math>x_2+x_3+x_4+x_6 \leq 3</math>		<math>x_2+x_3+x_4+x_6 \leq 3</math>
	== Flow Cover Cut==		== Flow Cover Cut==
	Flow Cover Cuts were introduced and derivced by Padberg, Van Roy, and Wolsey ~~[4]~~		Flow Cover Cuts were introduced and derivced by Padberg, Van Roy, and Wolsey(7)

	Given the system		Given the system
	<math> X = \{(x,y) \in R_+^n \times B^n : \sum{j \in N+} x_j - \sum_{j \in N-} x_j \leq d, x_j \leq m_jy_j, j \in N \} </math>		<math> X = \{(x,y) \in R_+^n \times B^n : \sum{j \in N+} x_j - \sum_{j \in N-} x_j \leq d, x_j \leq m_jy_j, j \in N \} </math>

2020-12-14T02:01:35Z

Flow Cover Cut

← Older revision		Revision as of 22:01, 13 December 2020
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	<math>x_2+x_3+x_4+x_6 \leq 3</math>		<math>x_2+x_3+x_4+x_6 \leq 3</math>
	== Flow Cover Cut==		== Flow Cover Cut==
			Flow Cover Cuts were introduced and derivced by Padberg, Van Roy, and Wolsey [4]
			Given the system
			<math> X = \{(x,y) \in R_+^n \times B^n : \sum{j \in N+} x_j - \sum_{j \in N-} x_j \leq d, x_j \leq m_jy_j, j \in N \} </math>

			where <math> N = N+ \cup N- </math> and <math> n = \|N\| </math>
			The flow cover inequality is defined as:

			<math> 0 \leq d - \sum_{j \in C+} x_j - \sum_{j \in C++}(m_j - l)(1-y_j) </math>

	~~<math>~~
	~~0 \leq d - \sum_{j \in C+} x_j - \sum_{j \in C++}\(m_j - l)(1-y_j) </math>~~
	Where <math> l = \sum_{j \in C+} m_j - d > 0 </math> and		Where <math> l = \sum_{j \in C+} m_j - d > 0 </math> and
	<math> C++ = \{j \in C+ : m_j > l\} </math>		<math> C++ = \{j \in C+ : m_j > l\} </math>

2020-12-14T01:46:53Z

← Older revision		Revision as of 21:46, 13 December 2020
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	<math>x_2+x_3+x_4+x_6 \leq 3</math>		<math>x_2+x_3+x_4+x_6 \leq 3</math>
	== Flow Cover Cut==		== Flow Cover Cut==

	<math>		<math>
	0 \leq d - \sum_{j \in C+}x_j - \sum_{j \in C++}(m_j - λ(1-y_j) </math>		0 \leq d - \sum_{j \in C+} x_j - \sum_{j \in C++}\(m_j - l)(1-y_j) </math>
	Where <math> λ = \sum_{j \in C+} m_j - d > 0 <\math> and		Where <math> l = \sum_{j \in C+} m_j - d > 0 </math> and
	<math> C++ = \{j \in C+ : m_j > λ\} </math>		<math> C++ = \{j \in C+ : m_j > l\} </math>

	== Numerical Example==		== Numerical Example==