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963
FACETS OF THE COMPLEMENTARITY KNAPSACK POLYTOPE
, 2002
"... We present a polyhedral study of the complementarity knapsack problem. Traditionally, complementarity constraints are modeled by introducing auxiliary binary variables and additional constraints, and the model is tightened by introducing strong inequalities valid for the resulting MIP. We use an alt ..."
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Cited by 7 (1 self)
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inequalities. We show that unlike 0–1 knapsack polytopes, in which different facetdefining inequalities can be derived by fixing variables at 0 or 1, and then sequentially lifting cover inequalities valid for the projected polytope, any sequentially lifted cover inequality for the complementarity knapsack
Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization
 SIAM Journal on Optimization
, 1993
"... We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized to S ..."
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Cited by 557 (12 self)
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We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized to SDP. Next we present an interior point algorithm which converges to the optimal solution in polynomial time. The approach is a direct extension of Ye's projective method for linear programming. We also argue that most known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity also carrying over in a similar fashion. Finally we study the significance of these results in a variety of combinatorial optimization problems including the general 01 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k partite subgraph problem in graphs, and va...
On Facets of Knapsack Equality Polytopes
, 1997
"... The 0/1 knapsack equality polytope is, by definition, the convex hull of 0/1 solutions of a single linear equation. A special form of this polytope  where the defining linear equation has nonnegative integer coefficients and the number of variables having coefficient one exceeds the righthandsid ..."
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Cited by 1 (1 self)
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The 0/1 knapsack equality polytope is, by definition, the convex hull of 0/1 solutions of a single linear equation. A special form of this polytope  where the defining linear equation has nonnegative integer coefficients and the number of variables having coefficient one exceeds the right
Hilbert Bases and the Facets of Special Knapsack Polytopes
 Mathematics of Operations Research
, 1994
"... Let a set N of items, a capacity F 2 IN and weights a i 2 IN, i 2 N be given. The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy the inequality X i2N a i x i F: In this paper we present a linear description of the 0/1 knapsack polytope for the special case where a i 2 f ..."
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Cited by 9 (4 self)
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description, facets, Hilbert basis, knapsack polytope, knapsack problem, separation 1 Introduction and Notation Let a set N of items, a capacity F 2 IN and weights a i 2 IN, i 2 N be given. The problem considered in this paper is the special case of the 0/1 knapsack problem, P i2N a i x i F , x i 2 f0; 1g
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. ..."
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Cited by 32 (5 self)
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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.
Facets for the ultipl Knapsack Problem
"... In this paper we consider the multiple knapsack problem which is defined a follows: given a set TV of items with weights fa, i G N, a set M of knapsacks with capacities Fk, k G M, and a profit function c^, z G N,k G M; find an assignment of a subset of the set of items to the set of knapsacks that y ..."
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of inequalities and work out necessary and sufficient conditions under which the corresponding inequality defines a facet. Some of these conditions involve only properties of certain knapsack constraints and hence, apply to the generalized assignment polytope as well. The results presented here serve
Facets for the Multiple Knapsack Problem
, 1993
"... In this paper we consider the multiple knapsack problem which is defined as follows: given a set N of items with weights f i , i 2 N , a set M of knapsacks with capacities F k , k 2 M , and a profit function c ik ; i 2 N; k 2 M ; find an assignment of a subset of the set of items to the set of knaps ..."
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of inequalities and work out necessary and sufficient conditions under which the corresponding inequality defines a facet. Some of these conditions involve only properties of certain knapsack constraints, and hence, apply to the generalized assignment polytope as well. The results presented here serve
THE SUBMODULAR KNAPSACK POLYTOPE
 FORTHCOMING IN DISCRETE OPTIMIZATION
, 2009
"... The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 01 knapsack set. One motivation for studying the submodular knapsack polytope is to address 01 programming problems with uncertain coefficients. Under various as ..."
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Cited by 1 (1 self)
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The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 01 knapsack set. One motivation for studying the submodular knapsack polytope is to address 01 programming problems with uncertain coefficients. Under various
Branchandprice: Column generation for solving huge integer programs
 Oper. Res
, 1998
"... We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branchandbound tree. We present classes of models for which th ..."
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Cited by 348 (13 self)
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We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branchandbound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. Wethen discuss computational issues and implementation of column generation, branchandbound algorithms, including special branching rules and e cient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality. 1
Results 1  10
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