A strengthened Barvinok-Pataki bound on SDP rank

Im, JiYoung; Wolkowicz, Henry

doi:10.1016/j.orl.2021.09.004

Cited by 4 publications

(4 citation statements)

References 23 publications

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“…It is known that every facial reduction yields at least one constraint being redundant, see e.g., [5], [21,Lemma 2.7], and [27,Section 3.5]. For completeness we now include a short proof tailored to LP, see Lemma 2.2.…”

Section: Facial Reductionmentioning

confidence: 99%

“…Remark 3.6. The idea used in the proof of Theorem 3.5 is the same as the one presented in [21] for spectrahedron. In [21], the authors use Proposition 3.2 to strengthen the bound called the Barvinok-Pataki bound.…”

Section: A Geometric Proof Using Extreme Pointsmentioning

confidence: 99%

“…The idea used in the proof of Theorem 3.5 is the same as the one presented in [21] for spectrahedron. In [21], the authors use Proposition 3.2 to strengthen the bound called the Barvinok-Pataki bound. The bound is strengthened by the means of singularity degree that stems from the facial reduction algorithm [12,23,27].…”

Section: A Geometric Proof Using Extreme Pointsmentioning

confidence: 99%

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Revisiting Degeneracy, Strict Feasibility, Stability, in Linear Programming

Im¹,

Wolkowicz²

2022

Preprint

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Currently, the simplex method and the interior point method are indisputably the most popular algorithms for solving linear programs. Unlike general conic programs, linear programs with a finite optimal value do not require strict feasibility in order to establish strong duality. Hence strict feasibility is often less emphasized. In this note we discuss that the lack of strict feasibility necessarily causes difficulties in both simplex and interior point methods. In particular, the lack of strict feasibility implies that every basic feasible solution is degenerate. We achieve this using facial reduction and simple means of linear algebra. Furthermore, we emphasize that facial reduction involves two steps where the first guarantees strict feasibility, and the second recovers full-row rankness of the constraint matrix.

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Section: Facial Reductionmentioning

confidence: 99%

Section: A Geometric Proof Using Extreme Pointsmentioning

confidence: 99%

Section: A Geometric Proof Using Extreme Pointsmentioning

confidence: 99%

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Revisiting Degeneracy, Strict Feasibility, Stability, in Linear Programming

Im¹,

Wolkowicz²

2022

Preprint

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“…In [20], the bound is used to explain how the exponential size solutions arise in SDP. In [21], a strengthened Barvinok-Pataki bound on SDP rank is derived using singularity degree. The singularity degree of the linear image of a cone is investigated in [22].…”

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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Revisiting degeneracy, strict feasibility, stability, in linear programming

Wolkowicz

2023

European Journal of Operational Research

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A strengthened Barvinok-Pataki bound on SDP rank

Cited by 4 publications

References 23 publications

Revisiting Degeneracy, Strict Feasibility, Stability, in Linear Programming

Revisiting Degeneracy, Strict Feasibility, Stability, in Linear Programming

Facial reduction for the Shor SDP relaxation of QCQPs

Revisiting degeneracy, strict feasibility, stability, in linear programming

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