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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 36 (11 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.
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 35 (9 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...
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 33 (3 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
, 2004
"... 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 s ..."
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Cited by 33 (0 self)
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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.
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 25 (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.
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 24 (7 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.
Separation algorithms for 01 knapsack polytopes
, 2008
"... Valid inequalities for 01 knapsack polytopes often prove useful when tackling hard 01 Linear Programming problems. To use such inequalities effectively, one needs separation algorithms for them, i.e., routines for detecting when they are violated. We show that the separation problems for the soca ..."
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Cited by 17 (0 self)
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Valid inequalities for 01 knapsack polytopes often prove useful when tackling hard 01 Linear Programming problems. To use such inequalities effectively, one needs separation algorithms for them, i.e., routines for detecting when they are violated. We show that the separation problems for the socalled extended cover and weight inequalities can be solved exactly in O(nb) time and O((n + amax)b) time, respectively, where n is the number of items, b is the knapsack capacity and amax is the largest item weight. We also present fast and effective separation heuristics for the extended cover and lifted cover inequalities. Finally, we present a new exact separation algorithm for the 01 knapsack polytope itself, which is faster than existing methods. Extensive computational results are also given.
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 14 (1 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.
Cover and Pack Inequalities for (Mixed) Integer Programming
"... We review strong inequalities for fundamental knapsack relaxations of (mixed) integer programs. These relaxations are the 01 knapsack set, the mixed 01 knapsack set, the integer knapsack set, and the mixed integer knapsack set. Our aim is to give a common presentation of the inequalities based ..."
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Cited by 10 (3 self)
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We review strong inequalities for fundamental knapsack relaxations of (mixed) integer programs. These relaxations are the 01 knapsack set, the mixed 01 knapsack set, the integer knapsack set, and the mixed integer knapsack set. Our aim is to give a common presentation of the inequalities based on covers and packs and highlight the connections among them. The focus of the paper is on recent research on the use of superadditive functions for the analysis of knapsack polyhedra. We also