Unifying semidefinite and set-copositive relaxations of binary problems and randomization techniques

Lieder, Felix; Rad, Fatemeh Bani; Jarre, Florian

doi:10.1007/s10589-015-9731-y

Cited by 5 publications

(7 citation statements)

References 12 publications

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“…The column headed "+(12)" was obtained by adding both constraints (11) and (12) to SDP1, and the column headed "+(13)" was obtained by adding, in addition, constraints (13). Similarly, the column headed "+(15)" was obtained by adding both constraints (14) and (15) to SDP2, and the column headed "+(17)" was obtained by adding, in addition, constraints (17). We did not experiment with adding constraints (16) or (18), since they have no analogue for SDP1.…”

Section: Computational Resultsmentioning

confidence: 99%

“…Generally speaking, adding cutting planes usually yields some improvement in the upper bound, but at the cost of dramatically increased computing times. Going in the opposite direction, Lieder et al [17] recently proposed to weaken the theta function, as follows. Take SDP2 and replace the equations (8) with the single equation…”

Section: Variants Of the Theta Functionmentioning

confidence: 99%

“…It is apparent that the inequalities (21) are the source of the inequalities (12) and (13) for SDP1 and the inequalities (15) and (17) for SDP2. The inequalities (22) are the source of the inequalities (16) and (18) for SDP2, but, being inhomogeneous, do not yield any useful inequalities for SDP1.…”

Section: Strengthening the Function With Cutting Planesmentioning

confidence: 99%

“…More generally, suppose we strengthen SDP2 by adding an arbitrary collection of homogeneous valid inequalities, such as some or all of (14), (15) and (17). Let (x,X) be the optimal solution to the strengthened relaxation.…”

Section: Strengthening the Function With Cutting Planesmentioning

confidence: 99%

“…Recall that the weakened theta function of Lieder et al [17] is obtained by taking the constraints (8) in SDP2 and replacing them with the single weaker constraint (19). We will denote this bound by ϑ LRJ (G).…”

Section: The Weakened Theta Functionmentioning

confidence: 99%

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On the Lovász theta function and some variants

Galli

Letchford

2017

Discrete Optimization

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The Lovász theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovász and the other to Grötschel et al. The former SDP is often thought to be preferable computationally, since it has fewer variables and constraints. We derive some new results on these two equivalent SDPs. The surprising result is that, if we weaken the SDPs by aggregating constraints, or strengthen them by adding cutting planes, the equivalence breaks down. In particular, the Grötschel et al. scheme typically yields a stronger bound than the Lovász one

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Section: Computational Resultsmentioning

confidence: 99%

Section: Variants Of the Theta Functionmentioning

confidence: 99%

Section: Strengthening the Function With Cutting Planesmentioning

confidence: 99%

Section: Strengthening the Function With Cutting Planesmentioning

confidence: 99%

Section: The Weakened Theta Functionmentioning

confidence: 99%

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On the Lovász theta function and some variants

Galli

Letchford

2017

Discrete Optimization

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Set-completely-positive representations and cuts for the max-cut polytope and the unit modulus lifting

et al. 2019

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This paper considers a generalization of the "max-cut-polytope" conv{ x x T | x ∈ R n , |x k | = 1 for 1 ≤ k ≤ n} in the space of real symmetric n × n-matrices with all-one diagonal to a complex "unit modulus lifting" conv{x x * | x ∈ C n , |x k | = 1 for 1 ≤ k ≤ n} in the space of complex Hermitian n × n-matrices with all-one diagonal. The unit modulus lifting arises in applications such as digital communications and shares similar symmetry properties as the max-cut-polytope. Set-completely positive representations of both sets are derived and the relation of the complex unit modulus lifting to its semidefinite relaxation is investigated in dimensions 3 and 4. It is shown that the unit modulus lifting coincides with its semidefinite relaxation in dimension 3 but not in dimension 4. In dimension 4 a family of deep valid cuts for the unit modulus lifting is derived that could be used to strengthen the semidefinite relaxation. It turns out that the deep cuts are also implied by a second lifting that could be used alternatively. Numerical experiments are presented comparing the first lifting, the second lifting, and the unit modulus lifting for n = 4.

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Simplified semidefinite and completely positive relaxations

Lieder

2015

Operations Research Letters

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Unifying semidefinite and set-copositive relaxations of binary problems and randomization techniques

Cited by 5 publications

References 12 publications

On the Lovász theta function and some variants

On the Lovász theta function and some variants

Set-completely-positive representations and cuts for the max-cut polytope and the unit modulus lifting

Simplified semidefinite and completely positive relaxations

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