Solving quadratic (0,1)-problems by semidefinite programs and cutting planes

Helmberg, Christoph; Rendl, Franz

doi:10.1007/bf01580072

Cited by 165 publications

(149 citation statements)

References 32 publications

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“…Whereas semidefinite programming relaxations of max-cut and related combinatorial problems have been investigated extensively (e.g., [5][6][7][8]), research on conic mixed-integer programming is so far fairly limited. Çezik and Iyengar [9] describe Chvátal-Gomory and disjunctive cuts for conic integer programs.…”

Section: Introductionmentioning

confidence: 99%

A strong conic quadratic reformulation for machine-job assignment with controllable processing times

Aktürk

Atamtürk

Gürel

2009

Operations Research Letters

102

145

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Section: Introductionmentioning

confidence: 99%

A strong conic quadratic reformulation for machine-job assignment with controllable processing times

Aktürk

Atamtürk

Gürel

2009

Operations Research Letters

102

145

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“…We adapt the rules R1-R4 of [28] for max-cut in order to derive different choices for branching variables.…”

Section: Branching Rulesmentioning

confidence: 99%

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Anjos

Ghaddar²,

Hupp

et al. 2013

Facets of Combinatorial Optimization

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This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching. We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k = 3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solv- ing the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far.

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“…These problems are reformulated as linear programs with an extra nonconvex constraint of the form Y = xx ⊤ , where Y is an n×n matrix of auxiliary variables. Relaxing these constraints to Y −xx ⊤ 0, thereby allowing solutions such that xx ⊤ − Y 0, leads to a (convex) semidefinite program, which yields good lower bounds [31,49].…”

Section: Mathematical Programs With Equilibrium Constraints (Mpec)mentioning

confidence: 99%

Disjunctive Inequalities: Applications And Extensions

Belotti

Liberti

Lodi

et al. 2011

Wiley Encyclopedia of Operations Research and Management Science

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We survey some applications and extensions of disjunctive programming with special emphasis on recent developments. Specifically, after recalling the basic ingredients of disjunctive inequalities we report on recent results in the context of mixed integer linear programming. We then consider the application of disjunctive constraints both as modeling tool and cutting planes in mixed integer nonlinear programming. Finally, we discuss the application of disjunctions as branching conditions in enumerative algorithms, as opposed to the cutting approach.

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Solving quadratic (0,1)-problems by semidefinite programs and cutting planes

Cited by 165 publications

References 32 publications

A strong conic quadratic reformulation for machine-job assignment with controllable processing times

A strong conic quadratic reformulation for machine-job assignment with controllable processing times

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Disjunctive Inequalities: Applications And Extensions

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