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40
Solving liftandproject relaxations of binary integer programs
 SIAM Journal on Optimization
"... Abstract. We propose a method for optimizing the liftandproject relaxations of binary integer programs introduced by Lovász and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constrain ..."
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Cited by 22 (2 self)
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Abstract. We propose a method for optimizing the liftandproject relaxations of binary integer programs introduced by Lovász and Schrijver. In particular, we study both linear and semidefinite relaxations. The key idea is a restructuring of the relaxations, which isolates the complicating constraints and allows for a Lagrangian approach. We detail an enhanced subgradient method and discuss its efficient implementation. Computational results illustrate that our algorithm produces tight bounds more quickly than stateoftheart linear and semidefinite solvers.
Semidefinite programming heuristics for surface reconstruction ambiguities
 In ECCV
, 2008
"... Abstract. We consider the problem of reconstructing a smooth surface under constraints that have discrete ambiguities. These problems arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based ..."
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Cited by 14 (1 self)
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Abstract. We consider the problem of reconstructing a smooth surface under constraints that have discrete ambiguities. These problems arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based on semidefinite programming may reveal the shape of the surface. 1
A scalable collusionresistant multiwinner cognitive spectrum auction game
 IEEE Trans. Wireless Commun
, 2009
"... Abstract—Dynamic spectrum access (DSA), enabled by cognitive radio technologies, has become a promising approach to improve efficiency in spectrum utilization, and the spectrum auction is one important DSA approach, in which secondary users lease some unused bands from primary users. However, spectr ..."
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Cited by 14 (7 self)
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Abstract—Dynamic spectrum access (DSA), enabled by cognitive radio technologies, has become a promising approach to improve efficiency in spectrum utilization, and the spectrum auction is one important DSA approach, in which secondary users lease some unused bands from primary users. However, spectrum auctions are different from existing auctions studied by economists, because spectrum resources are interferencelimited rather than quantitylimited, and it is possible to award one band to multiple secondary users with negligible mutual interference. To accommodate this special feature in wireless communications, in this paper, we present a novel multiwinner spectrum auction game not existing in auction literature. As secondary users may be selfish in nature and tend to be dishonest in pursuit of higher profits, we develop effective mechanisms to suppress their dishonest/collusive behaviors when secondary users distort their valuations about spectrum resources and interference relationships. Moreover, in order to make the proposed game scalable when the size of problem grows, the semidefinite programming (SDP) relaxation is applied to reduce the complexity significantly. Finally, simulation results are presented to evaluate the proposed auction mechanisms, and demonstrate the complexity reduction as well. Index Terms—Cognitive radio, spectrum auction, collusionresistant mechanism, scalable algorithm. I.
Semidefinite Programming Relaxations and Algebraic Optimization in Control
, 2003
"... We present an overview of the essential elements of semide nite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certi cates. Our focus is on the exciting d ..."
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Cited by 12 (4 self)
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We present an overview of the essential elements of semide nite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certi cates. Our focus is on the exciting developments occurred in the last few years, including robust optimization, combinatorial optimization, and algebraic methods such as sumofsquares. These developments are illustrated with examples of applications to control systems.
Cutting plane methods for semidefinite programming
, 2002
"... A semidefinite programming problem is a nonsmooth optimization problem, so it can be solved using a cutting plane approach. In this paper, we analyze properties of such an algorithm. We discuss characteristics of good polyhedral representations for the semidefinite program. We show that the complexi ..."
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Cited by 11 (5 self)
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A semidefinite programming problem is a nonsmooth optimization problem, so it can be solved using a cutting plane approach. In this paper, we analyze properties of such an algorithm. We discuss characteristics of good polyhedral representations for the semidefinite program. We show that the complexity of an interior point cutting plane approach based on a semiinfinite formulation of the semidefinite program has complexity comparable with that of a direct interior point solver. We show that cutting planes can always be found efficiently that support the feasible region. Further, we characterize the supporting hyperplanes that give high dimensional tangent planes, and show how such supporting hyperplanes can be found efficiently.
Multiparty quantum coin flipping
, 2003
"... We investigate coinflipping protocols for multiple parties in a quantum broadcast setting: (1) We propose and motivate a definition for quantum broadcast. (2) We prove that in this model, there are protocols for k parties with guaranteed bias 1/2 − Ω(1/k 1.78) for (k − 1) computationallyunbounded ..."
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Cited by 11 (1 self)
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We investigate coinflipping protocols for multiple parties in a quantum broadcast setting: (1) We propose and motivate a definition for quantum broadcast. (2) We prove that in this model, there are protocols for k parties with guaranteed bias 1/2 − Ω(1/k 1.78) for (k − 1) computationallyunbounded cheating parties. Classically, no bias < 1/2 can be achieved if a majority of the players is bad. (3) We extend our protocol to a setting where at most (1 −ǫ)k of the players are bad, for ǫ> 0. We show that in this case we can achieve a constant bias δ < 1/2 − Ω(ǫ 1.78), independent of k. (4) We show that our results are essentially optimal: for k parties with one good party, any quantum protocol can achieve bias at best 1/2 − O(1/k), while for (1 − ǫ)k cheating parties achieving constant bias δ < 1/2 − O(ǫ) is impossible, too. 1
THE OPERATOR Ψ FOR THE CHROMATIC NUMBER OF A GRAPH
, 2008
"... We investigate hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. We introduce an operator Ψ mapping any graph parameter β(G), nested between the stability number α(G) and χ(G), to a new graph parameter Ψβ(G), nested between α(G) and χ(G); Ψβ(G) is polynomial ti ..."
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Cited by 11 (1 self)
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We investigate hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. We introduce an operator Ψ mapping any graph parameter β(G), nested between the stability number α(G) and χ(G), to a new graph parameter Ψβ(G), nested between α(G) and χ(G); Ψβ(G) is polynomial time computable if β(G) is. As an application, there is no polynomial time computable graph parameter nested between the fractional chromatic number χ ∗ (·) and χ(·) unless P = NP. Moreover, based on the Motzkin–Straus formulation for α(G), we give (quadratically constrained) quadratic and copositive programming formulations for χ(G). Under some mild assumptions, n/β(G) ≤ Ψβ(G), but, while n/β(G) remains below χ ∗ (G), Ψβ(G) can reach χ(G) (e.g., for β(·) =α(·)). We also define new polynomial time computable lower bounds for χ(G), improving the classic Lovász theta number (and its strengthenings obtained by adding nonnegativity and triangle inequalities); experimental results on Hamming graphs, Kneser graphs, and DIMACS benchmark graphs will be given in the followup paper [N. Gvozdenović and M. Laurent, SIAM J. Optim., 19 (2008), pp. 592–615].
Expressing Combinatorial Optimization Problems by Systems of Polynomial Equations and the Nullstellensatz
, 2007
"... Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. In the first part of ..."
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Cited by 9 (5 self)
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Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. In the first part of this paper, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edgechromatic number, or the largest kcolorable subgraph. For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows. We show that the minimumdegree of a Nullstellensatz certificate for the nonexistence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non3colorability, we found only graphs with Nullstellensatz certificates of degree four.
Semidefinite Relaxations for MaxCut
 The Sharpest Cut, Festschrift in Honor of M. Padberg's 60th Birthday. SIAM
, 2001
"... We compare several semidefinite relaxations for the cut polytope obtained by applying the lift and project methods of Lov'asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the maxcut problem. This re ..."
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Cited by 9 (1 self)
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We compare several semidefinite relaxations for the cut polytope obtained by applying the lift and project methods of Lov'asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the maxcut problem. This relaxation Q t (G) can be defined as the projection on the edge subspace of the set F t (n), which consists of the matrices indexed by all subsets of [1; n] of cardinality t + 1 with the same parity as t + 1 and having the property that their (I ; J)th entry depends only on the symmetric difference of the sets I and J . The set F 0 (n) is the basic semidefinite relaxation of maxcut consisting of the semidefinite matrices of order n with an all ones diagonal, while Fn\Gamma2 (n) is the 2 n\Gamma1 dimensional simplex with the cut matrices as vertices. We show the following geometric properties: If Y 2 F t (n) has rank t + 1, then Y can be written as a convex combination of at most 2 t cut matrices, extending a result of Anjos and Wolkowicz for the case t = 1; any 2 t+1 cut matrices form a face of F t (n) for t = 0; 1; n \Gamma 2. The class L t of the graphs G for which Q t (G) is the cut polytope of G is shown to be closed under taking minors. The graph K 7 is a forbidden minor for membership in L 2 , while K 3 and K 5 are the only minimal forbidden minors for the classes L 0 and L 1 , respectively. 1