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Nonconvex Quadratic Programming: Return of the Boolean Quadric Polytope
"... We consider a nonconvex quadratic programming problem of the form: QP: min cTx + xTQx s.t. x ∈ B ∩ C. • B = {x  0 ≤ xi ≤ 1, i = 1,..., n}. • C is given by additional linear or quadratic constraints. We consider a nonconvex quadratic programming problem of the form: QP: min cTx + xTQx s.t. x ∈ B ∩ C ..."
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literature. Polyhedral approach to BQP, introduced by Padberg (1989), is based on studying the Boolean Quadric Polytope BQPn = conv{xi, yij  yij = xixj, 1 ≤ i < j ≤ n, xi ∈ {0, 1}, i = 1,..., n}. QPB is already NPhard. In particular, by adding an objective term γ ∑ni=1(xi−x2i) can represent
Applications of Cut Polyhedra
, 1992
"... We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole probl ..."
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Cited by 23 (2 self)
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We group in this paper, within a unified framework, many applications of the following polyhedra: cut, boolean quadric, hypermetric and metric polyhedra. We treat, in particular, the following applications: ffl ` 1  and L 1 metrics in functional analysis, ffl the maxcut problem, the Boole
Binary Positive Semidefinite Matrices and Associated Integer Polytopes
"... We consider the positive semidefinite (psd) matrices with binary entries. We give a characterisation of such matrices, along with a graphical representation. We then move on to consider the associated integer polytopes. Several important and wellknown integer polytopes — the cut, boolean quadric, ..."
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Cited by 3 (0 self)
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We consider the positive semidefinite (psd) matrices with binary entries. We give a characterisation of such matrices, along with a graphical representation. We then move on to consider the associated integer polytopes. Several important and wellknown integer polytopes — the cut, boolean quadric
On Generalized Network Design Polyhedra
, 2009
"... In recent years, there has been an increased literature on socalled Generalized Network Design Problems (GNDPs), such as the Generalized Minimum Spanning Tree Problem and the Generalized Traveling Salesman Problem. In a GNDP, the node set of a graph is partitioned into clusters, and the feasible ..."
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some interesting connections to other polyhedra, such as the quadratic semiassignment polytope and the boolean quadric polytope.
On nonconvex quadratic programming with box constraints
 SIAM J. on Optimiz
"... NonConvex Quadratic Programming with Box Constraints is a fundamental N Phard global optimisation problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dim ..."
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Cited by 19 (2 self)
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their dimension, characterise their extreme points and vertices, show their invariance under certain affine transformations, and show that various linear inequalities induce facets. We also show that the sets are closely related to the Boolean quadric polytope, a fundamental polytope in the field of polyhedral
The EdgeWeighted Clique problem: valid inequalities, facets and polyhedral computations
, 1997
"... Let Kn = (V; E) be the complete undirected graph with weights c e associated to the edges in E. We consider the problem of finding the subclique C = (U; F ) of Kn such that the sum of the weights of the edges in F is maximized and jU j b, for some b 2 [1; : : : ; n]. This problem is called the Maxi ..."
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Cited by 6 (1 self)
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of these inequalities. All instances with up to 48 nodes could be solved without entering into the branching phase. Moreover, we show that some of these new inequalities also define facets of the Boolean Quadric Polytope and generalize previously known inequalities for this polytope.
A Lagrangian Relaxation Approach to the EdgeWeighted Clique Problem
 European Journal of Operational Research
, 1999
"... The bclique polytope CP n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QP n = CP n n as a special case and being closely related to the quadratic knapsack polytope, it has recei ..."
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Cited by 16 (0 self)
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The bclique polytope CP n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QP n = CP n n as a special case and being closely related to the quadratic knapsack polytope, it has
unknown title
"... In this talk I will: • Describe some recent interesting results, • Say nice things about Laurence. In this talk I will: • Describe some recent interesting results, • Say nice things about Laurence. In this talk I will not: • Talk about our joint research. We consider a nonconvex quadratic programmin ..."
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, introduced by Padberg (1989), is based on studying the Boolean Quadric Polytope BQPn = conv{xi, yij  yij = xixj, 1 ≤ i < j ≤ n, xi ∈ {0, 1}, i = 1,..., n}. QPB is already NPhard. In particular, by adding an objective term γ ∑ni=1(xi−x2i) can represent the Boolean Quadratic Program BQP: min cTx + xTQx s
An Analysis of the Asymmetric Quadratic Traveling Salesman Polytope
 SIAM J. Discret. Math
"... Abstract. In the quadratic traveling salesman problem a cost is associated with any three nodes traversed in succession. This structure arises, e. g., if the succession of two edges represents energetic conformations, a change of direction or a possible change of transportation means. In the symmet ..."
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Cited by 3 (1 self)
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but offers a generic path towards proving facetness of several classes of valid inequalities. We establish relations to facets of the boolean quadric polytope, exhibit new classes of polynomial time separable facet defining inequalities that exclude conflicting configurations of edges, and provide a generic
Robustness In Solid Modeling  A Tolerance Based Intuitionistic Approach
, 1993
"... This paper presents a new robustness method for geometric modeling operations. Geometric relations are computed from the tolerances defined for geometric objects and the tolerances are dynamically updated to preserve the theoretical properties of the relations. The method is based on an intuitionist ..."
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Cited by 2 (1 self)
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on an intuitionistic selfvalidation approach. Geometric algorithms using this method are proved to be robust. We demonstrate the application of the approach for an algorithm computing Boolean set operations on objects bound by planar and natural quadric surfaces. A major advantage of the approach is that it can
Results 1  10
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