Improving the efficiency of the Branch and Bound algorithm for integer programming based on “flatness” information

Dérpich, Iván; Vera, Jorge

doi:10.1016/j.ejor.2005.02.051

Cited by 13 publications

(17 citation statements)

References 20 publications

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“…In their work, the branching rule utilizes the eigenvalues of a matrix corresponding to an inscribed Dikin ellipsoid [2]. However, experimenting with the method in [1] shows that is not adequately efficient as it requires long CPU processing times in calculating the center point of the ellipsoid and the eigenvalues of the matrix. The new proposed method uses the columns of matrix A.…”

Section: The Proposed Branching Rule: Flatness IImentioning

confidence: 99%

“…viii) The flatness rule (FLAT) is the original rule developed in [1]. Here, ellipses based on a logarithmic barrier function are used for the polyhedron.…”

Section: Branching Rulesmentioning

confidence: 99%

“…The rule is called Flatness II, named after the work by Derpich and Vera [1]. In their work, the branching rule utilizes the eigenvalues of a matrix corresponding to an inscribed Dikin ellipsoid [2].…”

Section: The Proposed Branching Rule: Flatness IImentioning

confidence: 99%

“…The basis of the rule is a minimum integer width vector. In a non-empty closed polyhedron K ⊂ the width of this vector d ∈ is given by [1]:…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Accelerating the B&B algorithm for integer programming based on flatness information: an approach applied to the multidimensional knapsack problem

Dérpich

Sepúlveda

2017

CRORR

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Abstract. This paper presents a new branching rule based on the flatness of a polyhedron associated to the set of constraints in an integer linear programming problem. The rule called Flatness II is a heuristic technique used with the branch-and-bound method. The rule is concerned with the minimum integer width vector. Empirical evidence supports the conjecture that the direction with the highest value of the vector's components indicates a suitable branching direction. The paper provides theoretical results demonstrating that the columns of the matrix A corresponding to a set of constraints Ax≤b may be used to estimate the minimum integer width vector; this fact is used for constructing a new version of the branching rule as was reported in a previous paper by the authors. In addition, the new rule uses a branching direction that chooses the child node closest to the integer value (either up or down). Thus, it uses a variable rule for descending the tree. Every time a new sub-problem is solved, the list of remaining unsolved sub-problems is analyzed, with priority given to those problems with a minimum objective function value estimate. The conclusions of the work are based on knapsack problems from the knapsack OR-Library. From the results, it is concluded that the new rule Flatness II presents low execution times and minimal number of nodes generated.

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Section: The Proposed Branching Rule: Flatness IImentioning

confidence: 99%

“…viii) The flatness rule (FLAT) is the original rule developed in [1]. Here, ellipses based on a logarithmic barrier function are used for the polyhedron.…”

Section: Branching Rulesmentioning

confidence: 99%

Section: The Proposed Branching Rule: Flatness IImentioning

confidence: 99%

“…The basis of the rule is a minimum integer width vector. In a non-empty closed polyhedron K ⊂ the width of this vector d ∈ is given by [1]:…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Accelerating the B&B algorithm for integer programming based on flatness information: an approach applied to the multidimensional knapsack problem

Dérpich

Sepúlveda

2017

CRORR

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“…It is also known, from a result of Banaszczyk et al [1999], that if P ∩ Z n is empty, then w(P) ≤ Cn 3 2 , where C is a constant. Derpich and Vera [2006] approximate the direction of the minimum width in order to assign priorities for branching on variables and use this to select variable disjunctions.…”

Section: Minimizing Widthmentioning

confidence: 99%

On the Complexity of Selecting Disjunctions in Integer Programming

Mahajan¹,

Ralphs²

2010

SIAM J. Optim.

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The imposition of general disjunctions of the form "πx ≤ π 0 ∨ πx ≥ π 0 + 1", where π, π 0 are integer valued, is a fundamental operation in both the branch-and-bound and cuttingplane algorithms for solving mixed integer linear programs. Such disjunctions can be used for branching at each iteration of the branch-and-bound algorithm or to generate split inequalities for the cutting-plane algorithm. We first consider the problem of selecting a general disjunction and show that the problem of selecting an optimal such disjunction, according to specific criteria described herein, is N P-hard. We further show that the problem remains N P-hard even for binary programs or when considering certain restricted classes of disjunctions. We observe that the problem of deciding whether a given inequality is a split inequality can be reduced to one of the above problems, which leads to a proof that the problem is N P-complete.

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Experiments with Branching using General Disjunctions

Mahajan

Ralphs

2009

Operations Research and Cyber-Infrastructure

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Branching is an important component of the branch-and-cut algorithm for solving mixed integer linear programs. Most solvers branch by imposing a disjunction of the form"x i ≤ k ∨ x i ≥ k + 1" for some integer k and some integer-constrained variable x i . A generalization of this branching scheme is to branch by imposing a more general disjunction of the form "πx ≤ π 0 ∨ πx ≥ π 0 + 1." In this paper, we discuss the formulation of two optimization models for selecting such a branching disjunction and then describe methods of solution using a standard MILP solver. We report on computational experiments carried out to study the effects of branching on such disjunctions.

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Improving the efficiency of the Branch and Bound algorithm for integer programming based on “flatness” information

Cited by 13 publications

References 20 publications

Accelerating the B&B algorithm for integer programming based on flatness information: an approach applied to the multidimensional knapsack problem

Accelerating the B&B algorithm for integer programming based on flatness information: an approach applied to the multidimensional knapsack problem

On the Complexity of Selecting Disjunctions in Integer Programming

Experiments with Branching using General Disjunctions

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