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23
Lifted Flow Cover Inequalities for Mixed 01 Integer Programs
 Mathematical Programming
, 1996
"... We investigate strong inequalities for mixed 01 integer programs derived from flow cover inequalities. Flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities. However, because of the sequential nature of the standard lifting techniques and the c ..."
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Cited by 34 (8 self)
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We investigate strong inequalities for mixed 01 integer programs derived from flow cover inequalities. Flow cover inequalities are usually not facet defining and need to be lifted to obtain stronger inequalities. However, because of the sequential nature of the standard lifting techniques and the complexity of the optimization problems that have to be solved to obtain lifting coefficients, lifting of flow cover inequalities is computationally very demanding. We present a computationally efficient way to lift flow cover inequalities based on sequence independent lifting techniques and computational results that justify the effectiveness of our lifting procedures. 1 Introduction A mixed integer program (MIP) with binary integer variables (BMIP) is the appropriate mathematical model for many practical optimization problems. This model is used, for example, for facility location problems, distribution problems, network design problems and more generally when fixed or concave costs are re...
FATCOP: A fault tolerant CondorPVM mixed integer program solver. Mathematical Programming
, 1999
"... Abstract. We describe FATCOP, a new parallel mixed integer program solver written in PVM. The implementation uses the Condor resource management system to provide a virtual machine composed of otherwise idle computers. The solver differs from previous parallel branchandbound codes by implementing ..."
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Cited by 27 (4 self)
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Abstract. We describe FATCOP, a new parallel mixed integer program solver written in PVM. The implementation uses the Condor resource management system to provide a virtual machine composed of otherwise idle computers. The solver differs from previous parallel branchandbound codes by implementing a general purpose parallel mixed integer programming algorithm in an opportunistic multiple processor environment, as opposed to a conventional dedicated environment. It shows how to make effective use of resources as they become available while ensuring the program tolerates resource retreat. The solver performs well on test problems arising from real applications and is particularly useful for solving long running hard mixed integer programming problems.
On the facets of the mixed–integer knapsack polyhedron
 MATH. PROGRAM., SER. B 98: 145–175 (2003)
, 2003
"... We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities cont ..."
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Cited by 27 (12 self)
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We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed–integer programming.
Sequence Independent Lifting in Mixed Integer Programming
 Journal of Combinatorial Optimization
, 1998
"... We investigate lifting, i.e., the process of taking a valid inequality for a polyhedron and extending it to a valid inequality in a higher dimensional space. Lifting is usually applied sequentially, that is, variables in a set are lifted one after the other. This may be computationally unattractive ..."
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Cited by 25 (2 self)
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We investigate lifting, i.e., the process of taking a valid inequality for a polyhedron and extending it to a valid inequality in a higher dimensional space. Lifting is usually applied sequentially, that is, variables in a set are lifted one after the other. This may be computationally unattractive since it involves the solution of an optimization problem to compute a lifting coefficient for each variable. To relieve this computational burden, we study sequence independent lifting, which only involves the solution of one optimization problem. We show that if a certain lifting function is superadditive, then the lifting coefficients are independent of the lifting sequence. We introduce the idea of valid superadditive lifting functions to obtain good aproximations to maximum lifting. We apply these results to strengthen Balas' lifting theorem for cover inequalities and to produce lifted flow cover inequalities for a single node flow problem. December 1995 Revised July 1997 Revised Janua...
Integerprogramming software systems
 Annals of Operations Research
, 1995
"... Abstract. Recent developments in integer–programming software systems have tremendously improved our ability to solve large–scale instances. We review the major algorithmic components of state–of–the–art solvers and discuss the options available to users to adjust the behavior of these solvers when ..."
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Cited by 24 (0 self)
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Abstract. Recent developments in integer–programming software systems have tremendously improved our ability to solve large–scale instances. We review the major algorithmic components of state–of–the–art solvers and discuss the options available to users to adjust the behavior of these solvers when default settings do not achieve the desired performance level. Furthermore, we highlight advances towards integrated modeling and solution environments. We conclude with a discussion of model characteristics and substructures that pose challenges for integer–programming software systems and a perspective on features we may expect to see in these systems in the near future. 1.
Exact solutions to linear programming problems
 Operations Research Letters
, 2007
"... The use of floatingpoint calculations limits the accuracy of solutions obtained by standard LP software. We present a simplexbased algorithm that returns exact rational solutions, taking advantage of the speed of floatingpoint calculations and attempting to minimize the operations performed in ra ..."
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Cited by 19 (8 self)
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The use of floatingpoint calculations limits the accuracy of solutions obtained by standard LP software. We present a simplexbased algorithm that returns exact rational solutions, taking advantage of the speed of floatingpoint calculations and attempting to minimize the operations performed in rational arithmetic. Extensive computational results are presented.
The median cycle problem
, 2001
"... In the Median Cycle Problem the aim is to determine a simple cycle containing a subset of vertices of a graph, while considering two types of cost: routing costs associated with the cycle itself, and the cost of assigning vertices not on the cycle to vertices of the cycle. Two versions of the proble ..."
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Cited by 8 (0 self)
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In the Median Cycle Problem the aim is to determine a simple cycle containing a subset of vertices of a graph, while considering two types of cost: routing costs associated with the cycle itself, and the cost of assigning vertices not on the cycle to vertices of the cycle. Two versions of the problem are investigated. In the rst, the objective function is the sum of routing and assignment costs. In the second, routing costs are minimized, subject to an upper bound on the total assignment cost. The two versions are formulated as integer linear programs. The polyhedral structure of the rst model is analyzed and the second model is strengthened through the introduction of additional valid inequalities. Separation procedures are developed. Heuristic procedures and an exact branchandcut algorithm are described. Computational results on benchmark instances and on randomly generated problems con rm the e ciency of the proposed algorithms.
Locating median cycles in networks
, 2005
"... In the median cycle problem the aim is to determine a simple cycle through a subset of vertices of a graph involving two types of costs: a routing cost associated with the cycle itself, and the cost of assigning vertices not on the cycle to visited vertices. The objective is to minimize the routing ..."
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Cited by 8 (0 self)
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In the median cycle problem the aim is to determine a simple cycle through a subset of vertices of a graph involving two types of costs: a routing cost associated with the cycle itself, and the cost of assigning vertices not on the cycle to visited vertices. The objective is to minimize the routing cost, subject to an upper bound on the total assignment cost. This problem arises in the location of a circularshaped transportation and telecommunication infrastructure. We present a mixed integer linear model, and strengthen it with the introduction of additional classes of nontrivial valid inequalities. Separation procedures are developed and an exact branchandcut algorithm is described. Computational results on instances from the classical TSP library and randomly generated ones confirm the efficiency of the proposed algorithm. An application related to the city of Milan (Italy) is also solved within reasonable computation time.
Primal cutting plane algorithms revisited
 MATH METH OPER RES (2002) 56:67–81
, 2002
"... Dual fractional cutting plane algorithms, in which cutting planes are used to iteratively tighten a linear relaxation of an integer program, are wellknown and form the basis of the highly successful branchandcut method. It is rather less wellknown that various primal cutting plane algorithms wer ..."
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Cited by 7 (2 self)
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Dual fractional cutting plane algorithms, in which cutting planes are used to iteratively tighten a linear relaxation of an integer program, are wellknown and form the basis of the highly successful branchandcut method. It is rather less wellknown that various primal cutting plane algorithms were developed in the 1960s, for example by Young. In a primal algorithm, the main role of the cutting planes is to enable a feasible solution to the original problem to be improved. Research on these algorithms has been almost nonexistent. In this paper we argue for a reexamination of these primal methods. We describe a new primal algorithm for pure 01 problems based on strong valid inequalities and give some encouraging computational results. Possible extensions to the case of general mixedinteger programs are also discussed.