The Many Faces of Degeneracy in Conic Optimization

Wolkowicz, Henry

doi:10.1561/2400000011

Cited by 33 publications

(6 citation statements)

References 107 publications

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“…If one can find all such |v , one can then perform facial reduction [6] on the SDP, reformulating it to lie on the orthogonal complement of these vectors, which will result in a smaller SDP that is necessarily strictly feasible. In fact, here we will not need to perform full facial reduction.…”

Section: Reformulating Sdpsmentioning

confidence: 99%

“…The crucial question is how to find these vectors |v . We'll use the Theorem of the Alternative for that [6]:…”

Section: Reformulating Sdpsmentioning

confidence: 99%

“…Let's now turn to the problem of maximizing µ under the constraint that l 1 (µ), defined in Equation ( 1), belongs to Almost Quantum. The resulting SDP, considered again as the dual form (6), is max µ c 01,10 c 01,1…”

Section: A First Problemmentioning

confidence: 99%

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Comment on “Geometry of the quantum set on no-signaling faces”

Araújo

2023

Phys. Rev. A

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Section: Reformulating Sdpsmentioning

confidence: 99%

“…The crucial question is how to find these vectors |v . We'll use the Theorem of the Alternative for that [6]:…”

Section: Reformulating Sdpsmentioning

confidence: 99%

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Comment on “Geometry of the quantum set on no-signaling faces”

Araújo

2023

Phys. Rev. A

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“…Through removing redundancy, FRA provides a robust framework for recovering a constraint qualification called Slater's condition. An excellent survey of the facial reduction algorithm can be found in [4].…”

Section: Introductionmentioning

confidence: 99%

Facial reduction for the Shor SDP relaxation of QCQPs

2023

Appl. Set-Valued Anal. Optim.

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We propose a special facial reduction algorithm (FRA) for the Shor semidefinite programming (SDP) relaxation of the quadratically constrained quadratic program (QCQP). Under the mild assumption, our special FRA only requires solving a linear programming problem instead of a semidefinite program. In particular, when applied to the binary quadratic program, the proposed special FRA needs fewer assumptions. From a computational perspective, this result improves the scalability and stability of the SDP approach for QCQP problems. In addition, we also discover a new class of semidefinite programs whose singularity degree can be computed easily. This new class complements the limited examples of singularity degrees in the literature. As a by-product, our special FRA can be used to upper bound the dimension of any set defined by quadratic inequalities.

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“…One of the most commonly used CQ is the Slater condition that consists in non-emptiness of the interior of the feasible set. For SDP problems that do not satisfy this CQ, several regularisation procedures based on a so-called facial reduction approach (FRA) were proposed in [6][7][8][9]. Being very general (they are designed for a wider class of problems than SDP), these procedures are not detailed.…”

Section: Introductionmentioning

confidence: 99%

Linear semidefinite programming problems: regularisation and strong dual formulations

Kostyukova

Tchemisova

2020

Journal of the Belarusian State University. Mathematics and Inf

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Regularisation consists in reducing a given optimisation problem to an equivalent form where certain regularity conditions, which guarantee the strong duality, are fulfilled. In this paper, for linear problems of semidefinite programming (SDP), we propose a regularisation procedure which is based on the concept of an immobile index set and its properties. This procedure is described in the form of a finite algorithm which converts any linear semidefinite problem to a form that satisfies the Slater condition. Using the properties of the immobile indices and the described regularization procedure, we obtained new dual SDP problems in implicit and explicit forms. It is proven that for the constructed dual problems and the original problem the strong duality property holds true.

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The Many Faces of Degeneracy in Conic Optimization

Cited by 33 publications

References 107 publications

Comment on “Geometry of the quantum set on no-signaling faces”

Comment on “Geometry of the quantum set on no-signaling faces”

Facial reduction for the Shor SDP relaxation of QCQPs

Linear semidefinite programming problems: regularisation and strong dual formulations

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