Results 1 
9 of
9
The Sequential Knapsack Polytope
 MATHEMATICAL PROGRAMMING
, 1998
"... In this paper we describe the convex hull of all solutions of the integer bounded knapsack problem in the special case when the weights of the items are divisible. The corresponding inequalities are defined via an inductive scheme that can also be used in a more general setting. ..."
Abstract

Cited by 32 (5 self)
 Add to MetaCart
In this paper we describe the convex hull of all solutions of the integer bounded knapsack problem in the special case when the weights of the items are divisible. The corresponding inequalities are defined via an inductive scheme that can also be used in a more general setting.
Cutting Planes for Integer Programs with General Integer Variables
, 1995
"... We investigate the use of cutting planes for integer programs with general integer variables. We show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 01 variables. We also explore the use of Gomory's mixed integer cuts. We address both theore ..."
Abstract

Cited by 22 (7 self)
 Add to MetaCart
We investigate the use of cutting planes for integer programs with general integer variables. We show how cutting planes arising from knapsack inequalities can be generated and lifted as in the case of 01 variables. We also explore the use of Gomory's mixed integer cuts. We address both theoretical and computational issues and show how to embed these cutting planes in a branchandbound framework. We compare results obtained by using our cut generation routines in two existing systems with a commercially available branchandbound code on a range of test problems arising from practical applications.
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 ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
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
The height of minimal Hilbert bases
"... For an integral polyhedral cone C = pos{a1,..., am}, ai ∈ Zd, a subset B(C) ⊂ C ∩ Zd is called a minimal Hilbert basis of C iff (i) each element of C∩Zd can be written as a nonnegative integral combination of elements of B(C) and (ii) B(C) has minimal cardinality with respect to all subsets of C ∩ ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
For an integral polyhedral cone C = pos{a1,..., am}, ai ∈ Zd, a subset B(C) ⊂ C ∩ Zd is called a minimal Hilbert basis of C iff (i) each element of C∩Zd can be written as a nonnegative integral combination of elements of B(C) and (ii) B(C) has minimal cardinality with respect to all subsets of C ∩ Zd for which (i) holds. We give a tight bound for the socalled height of an element of the basis which improves on former results.
Intermediate integer programming representations using value disjunctions
 Discrete Optimization
"... ..."
(Show Context)
Strong formulations for mixed integer programs: valid inequalities and extended formulations
 MATHEMATICAL PROGRAMMING B
, 2003
"... We examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the “easy” and “hard” sides of the complexity frontier. Most progress has been made in studying sets arising from knapsack and single node ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
We examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the “easy” and “hard” sides of the complexity frontier. Most progress has been made in studying sets arising from knapsack and single node flow sets, and a variety of sets motivated by different lotsizing models. We conclude by citing briefly some of the more intriguing new avenues of research.
Lifting 2integer knapsack inequalities
, 2003
"... In this paper we discuss the generation of strong valid inequalities for (mixed) integer knapsack sets based on lifting of valid inequalities for basic knapsack sets with two integer variables (and one continuous variable). The description of the basic polyhedra can be made in polynomial time. We us ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
In this paper we discuss the generation of strong valid inequalities for (mixed) integer knapsack sets based on lifting of valid inequalities for basic knapsack sets with two integer variables (and one continuous variable). The description of the basic polyhedra can be made in polynomial time. We use superadditive valid functions in order to obtain sequence independent lifting. 1
Cutting Planes and the Sequential Knapsack Problem
, 1994
"... Hartmann and Olmstead recently described an O(n log n) algorithm for solving sequential knapsack problems, and applied it to a relaxation of the 01 knapsack problem to compute a better bound than z LP in linear time after sorting the ratios p j =w j . This note extends these results in several dir ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Hartmann and Olmstead recently described an O(n log n) algorithm for solving sequential knapsack problems, and applied it to a relaxation of the 01 knapsack problem to compute a better bound than z LP in linear time after sorting the ratios p j =w j . This note extends these results in several directions. First we show how a similar bound can be obtained for the bounded knapsack problem. These bounds can be strengthened using Lagrangian relaxation, although the time required to solve the relaxation may increase by a factor of O(log n). In the 01 case, the optimal value of the Lagrangian dual can be interpreted as the result of adding a class of cutting planes to the linear programming relaxation of the 01 knapsack problem. We show that the separation problem for this class of cutting planes can be reduced to the separation problem over the bounded sequential knapsack polytope. We give a complete description of the bounded sequential knapsack problem in two special cases, and show...